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I wish the bastards dead," said Mark Rylance.

The audience laughed and gasped. It's not usually a funny line. In Act Four of Richard III, Shakespeare's psychopathic king is ordering the murder of two children. But when the instruction arrived clear as ice water out of Rylance's mouth, those five words - by some alchemy - became corrosively funny. A monstrous decree became banal, offhand. It was as if Richard were asking for the leaves to be cleared from his porch.

I stood yards away from Rylance that night, with my elbows on the stage of Shakespeare's Globe, transfixed. It was 31 July 2012.

The London Olympics were on and the theatre was swollen with tourists. Five-pound punters stood in the roofless yard in shirtsleeves, while helicopters buzzed overhead and church bells sounded on the hour. It had been seven years since Rylance had performed on this stage, with the company he led for a decade as artistic director; seven years since he had walked away from this building, dejected. Now here he was again, the greatest Shakespearean actor of his generation - maybe the greatest actor regardless of category - playing the title role inRichard III and Olivia in Twelfth Night in back-to-back productions.

When Rylance said the "bastards" line, he held more than 1,000 people in the palm of his hand. But he felt little pleasure, either in its delivery or in the reaction it generated. There was, instead, a sting in his chest.

The work of an actor, he would tell me two years later, "is to be totally present on stage at every moment, as if it's never happened before, with all your heart and mind and soul, anything you can muster". On that warm July night, he had mustered poison.

Four weeks previously, his stepdaughter Nataasha had died from a brain haemorrhage. She was 28, apparently healthy, deeply beloved.

Her funeral took place on the day before the Globe run began. It hurt Rylance to stand on stage and say these evil things, when what he needed to do was comfort his family and heal himself, but he also recalled that the experience was cathartic. "Acting has always been the cure for me. And the rage of Richard... the determination that nature and the world was reckless and cruel, and who gives a f***. If I can f*** you over, I will f*** you over. Why not? That was a strong feeling in me at the time. If nature can do this to me, then I don't care about anything or anyone."

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Rylance's performance that evening was raw, unforgettable and at times so hilarious it made your eyes water. As the play ended, the audience cheered and stomped. He received the acclaim with grace.

Then, as he would do every night of the Globe run, he retired backstage, took off his costume, spoke to few if any people, stepped on his bicycle and rode through the humming city back to his house on Shakespeare Road in south London.

"Mark Rylance speaks Shakespeare as if it was written for him the night before." (Al Pacino)

Mark Rylance is 54 years old and a peculiar sort of famous. Ask most actors or directors who they believe to be the finest actor working today, and they'll say his name. (Take Al Pacino: "Mark Rylance speaks Shakespeare as if it was written for him the night before." Or Steven Spielberg, who met him for the first time backstage after watching him in Twelfth Night and tells GQ: "I really think this actor can do anything." Or Bryan Cranston: "I have a huge man-crush.")

But he is of no interest to the gossip magazines and he spends most of his life happily unrecognised. Before taking his photograph for this magazine, David Bailey teased him by saying, "Who are you? A thespian?"

Part of the dissonance between Rylance's reputation and his public persona is straightforwardly explained: he has done most of his memorable work on the stage, rather than on-screen. But that's not the whole answer. Even though he's now warming up to the movies - he will shortly star in two Steven Spielberg pictures, The BFG and St James Place, as well as playing Thomas Cromwell in the BBC adaptation of Hilary Mantel's Wolf Hall - something in his appearance and temperament is unsuited to membership of the celebrity-industrial complex.

Meeting Rylance this summer at a café near his home in Herne Hill, where he remained unpestered over three hours, I was struck by how difficult he is to physically describe. He's handsome in a ragged way and a bit thin on top. But that does not adequately describe his effect. His accent wanders from Midwest to west Wiltshire. He laughs a great deal and at odd moments. He does a lot of work with his eyes. There are times when his mouth dances naughtily and he reminds you of a miscreant teenager, others when he seems as old and furrowed as a superannuated docker. Not many 50-something male actors could convincingly play a young woman, as Rylance has done recently, but you can see why it's no problem for him. His face seems to contain all the faces.

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There's also the problem of where to place Rylance intellectually. He's known as an eccentric. He forged his reputation playing Shakespeare, but is considered a heretic on the question of the authorship of Shakespeare's plays (even if his actual views on the "authorship question" are slightly more nuanced, as he will explain.) He makes key decisions in his life by throwing the Chinese "I Ching" divining dice. He attends conferences on crop circles.

If you only knew these things about Rylance, you might dismiss him as another luvvie flake, but even if you had only watched him for a few minutes on stage, you would understand that he must be more interesting than the caricature allows. His performance as the drug-pushing, hard-drinking Rooster Byron in Jez Butterworth's play Jerusalem remains many people's idea of the greatest lead role in recent stage history. It's certainly the strongest performance I've ever seen: profound, profane, deeply moving.

But watching him act in a bad or indifferent play is equally interesting. As Mike Alfreds, the director Rylance often cites as a mentor, tells me, "He's a chameleon. He's able to transform himself into almost anything. Everything he does, he takes into a space you don't expect. It's a sort of genius. He has enormous capacity and intense commitment, an animal vitality you can't take your eyes off."

A vapid or stupid person could not act as Rylance does. But he does seem to have a tolerance for unusual ideas and his character seems less fixed than you might expect of a man his age. How to pin him down? The crank and the great soul? The heretic and the genius?

To begin to answer those questions, you need to understand where he comes from. And where he comes from, improbably, is Milwaukee.

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David Mark Rylance Waters was born in Ashford, in Kent, in 1960, to two English teachers, Anne and David Waters.

The family moved to Connecticut in 1962 and then Wisconsin in 1969, where Rylance's parents worked at the University School of Milwaukee. Mark, who was the eldest of Anne and David's three children, was a slow developer. He couldn't speak in a way that anybody except his younger brother could understand until he was five years old. He had plenty of vowels but no consonants. As a result, during his first years at preschool he hardly said a word.

Rylance remembers that he conquered his impediment through playing "imaginary games" with his friends - acting, in other words. "There were terrific shows on TV like Star Trek and Voyage To The Bottom Of The Sea and Wild Wild West," he says. "All us kids would watch them. We would act them out in the basement. I think I found that I could speak a bit more clearly when I was playing with other people."

When the family moved to Milwaukee, this fondness for imaginary games turned into a full-blown affair. At high school, Rylance would not only act in every play but build the sets, fix the lights and write the programmes. He was not always a knockout. He remembers performing in a musical when he was 13 or 14. On the ride home, his parents talked about every other child in the play except for him and he was too nervous to ask his parents what they thought of his performance. Years later his father told him, "Sorry, we thought you were appalling. We didn't know what to say." Rylance laughs himself half to death at the memory of it.

He improved. In the summer holidays, his parents would return to England to teach at summer schools, with Rylance in tow. It was an opportunity to see plays at the Royal Shakespeare Company in Stratford and he was enraptured. In 1976, when Rylance was 16, he was cast as Hamlet - and he carried around a paperback copy of the play all summer to learn his lines. Marshall Sella, who was Rylance's best friend at school and is now a magazine writer, remembers that his performance impressed the teachers so much that they suggested the production should tour the east coast of America. "It was absolutely unheard of that they would let students just leave for three weeks or so," remembers Sella. "But there was this recognition that there was something very extraordinary going on with him and that play. High-school productions of Hamlet are mostly a slow death... But he said the words and people understood them. It was moving and interesting. The ideas were coming right into your brain."

"I had a lot of problems and the way I dealt with them was through acting." (Mark Rylance)

In December 1977, shortly before his 18th birthday, Rylance travelled to New York to audition for a visiting panel from The Royal Academy Of Dramatic Arts - Rada, the London acting school - because it was cheaper than most American acting colleges and less weird. "All those courses [in the US] were a couple of years of breaking you down psychologically," he remembers. "Not much acting.

It was to make you emotionally available, to get all your problems out. Now, I had a lot of problems and the way I dealt with them was through acting. Not going in front of people and them all criticising me until I cried. I was very nervous about that."

For his Rada audition, Rylance performed a monologue from Hamlet. The school took him - this Midwestern kid with the strangely convincing English accent - and he studied with a small class that included Sir Kenneth Branagh. His progress from then onwards was vertiginous: a year at the Glasgow Citizens Theatre with, among others, Gary Oldman. Then, via some detours, he arrived at The Royal Shakespeare Company, where he took lead role after lead role. The Guardian's Lyn Gardner wrote that in 1984, during the RSC's production of Peter Pan at the Barbican, in which Rylance played the title character, she saw the whole audience in tears.

The zenith of Rylance's work at the RSC came in 1989, when he played Hamlet as a mental patient in dirty pyjamas. Rylance remembers one performance in particular was unforgettable - a one-off at Broadmoor Hospital. There was one patient who told him, "You really were mad... I should know, I'm a loony." And then, as Rylance was changing after the show, a cigar-smoking figure he recognised approached him. It was Jimmy Savile, who told Rylance, "Good stuff, good stuff," (Rylance's Savile impression is spot on) and then gave him a Jim'll Fix It medal, which he still owns. "And now," says Rylance, "I want to know why Savile was there."

It was during this period that Rylance began to understand what he could achieve as an actor. The crossroads moment came in 1986, when Steven Spielberg offered him a role in Empire Of The Sun. Rylance initially turned him down. Spielberg called back, offering a bigger part. Rylance said yes. The money was around £15,000, he remembers - "enough to get a mortgage or something" - and Rylance was skint. But then the stage director Mike Alfreds called the same day, saying he was about to direct a season at the National Theatre and that Rylance could have "pretty much any part he wanted". Rylance was stumped. He loved working with Alfreds.

Spielberg was Spielberg. He let the I Ching divining dice decide - an odd solution, given that there are 64 possible outcomes every time you roll.

"You don't get a community on film sets. That solved is for me. I chose the theatre." (Mark Rylance) "You have to ask the dice a question," says Rylance. "The best question is, 'Where now?' So I asked, 'Where now if I do the Spielberg one?' and I threw. And I asked, 'Where now if I do the National Theatre?'" He can't remember what he rolled for the Spielberg film, but he remembers that the dice's answer for the National Theatre question was the hexagram meaning "community". "You don't get a community on film sets," he says. "The crew has a community, but the actors don't... That solved it for me. I chose the theatre."

The National Theatre season was a triumph. On that job, he also met his future wife, the composer Claire van Kampen - who had two young daughters, Juliet and Nataasha, both of whom would eventually call Rylance "Dad". The experience of making the choice between movies and the stage when the stakes appeared so high had also helped Rylance understand what kind of a career he wanted. "I realised that the reason I was in it, the reason I enjoyed it, was play-acting," he says. "Nothing else mattered. The money didn't matter. OK, it's nice to have enough to get by, but I wasn't in it for that. I wasn't in it to be notorious or famous."

The question is: why couldn't films make him happy before? And why - if not to become notorious or famous - does he think they can make him happy now?

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On a balmy day last June, at Broughton Castle in Oxfordshire, Rylance began to answer that question for me. He was midway through shooting Wolf Hall for the BBC, playing Henry VIII's scheming minister Thomas Cromwell. On the lawn outside the castle, crew members sat drinking coffee on collapsible chairs. Beside them was a props table, laden with glass grapes for the banqueting scenes. An extra in full Tudor costume refilled her water bottle at the cooler.

Inside the house, Rylance was shooting a 20-second scene by candlelight, in which Cromwell and Henry VIII - played by Damian Lewis - examine Anne Boleyn's bed, which had been set on fire. It was an unremarkable section of the narrative, one of 100 tiny modulations in a complex story. But something interesting happened when Rylance played it. Over dozens of takes, he gave the director, Peter Kosminsky, a slightly different performance every time. (Once, between takes, he played the whole thing as Lieutenant Columbo, to the amusement of his colleagues.) He couldn't bear for even this tiny moment to get boring for him or the other actors.

Kosminsky, looking into a monitor, purred at the end of each take, sometimes mumbling a quick "wonderful" or "interesting".

Often, he'd ask Rylance to try something different, but none of his instructions were delivered as fiats. He knew he was in good hands.

When the shoot wrapped, Kosminsky suggested that part of Rylance's greatness lay in the restlessness I had witnessed at Broughton Castle. "It is something I've thought about a lot," said Kosminsky. "Why is he different from other actors? He's quite uniquely vulnerable.

He's very open to the vibrations and emotions around him. He's very quick to laugh. He's quick to take offence. There's very little in the way of mask or suit of armour around him. It's as if he's on receive the whole time, rather than send. A lot of actors, you can't get them to shut up. Instead of listening and watching, they're always telling you something. Mark's the opposite."

Kosminsky is one of the few screen directors with whom Rylance has worked more than once. In 2004, they made The Government Inspector for Channel 4 together, about the death of the British weapons inspector, David Kelly. Rylance likes Kosminsky, a soft-spoken persuader. His experiences on other film sets have not been so happy.

In 1987, he played a minor role in Hearts Of Fire - a dire Bob Dylan vehicle. Rylance now says he only took the part because he adored Dylan and wanted to befriend him. (He still has dreams in which he is at his grandparents' house in Sissinghurst in Kent and aBlonde On Blonde-era Dylan is playing him his new songs and asking whether he likes them.

Sometimes Dylan also asks Rylance to cut his hair.) But the film and the friendship did not get far. It was at the time that Gorbachev and Reagan were meeting in Iceland and the Cold War appeared to be thawing. Rylance thought he'd engage the singer in some political chitchat. "I asked him, 'What do you think's happening in Reykjavik?'"

Rylance remembers, laughing so hard he can hardly tell the story. "And he looked at me so askance and said, 'I don't know nothing about nothing.' As I said the words, I saw them flying towards him, and I was thinking, 'No! Bob gave up politics in the Sixties! What are you doing? What are you doing?'"

His experience on Intimacy, a relationship drama directed by Patrice Chéreau in which the actors were told to play the sex scenes for real, was more seriously distressing. "I was very unhappy," says Rylance. "Patrice's style was very demanding of me. I lost confidence. His methodology was very critical. I don't like to be told what to do."

After each of these bombs, Rylance would ask himself why he subjected himself to the torture of making films. He didn't enjoy the process. He felt cut off from the audience. But he kept going back because of the pressure all actors feel to do television and film. It pays more. It makes you more famous. And with that fame and money, you can perhaps do more interesting projects on the stage. "I'd kind of worried about it my whole life, to be honest, a little bit," says Rylance. "I'd seen so many of my generation -

Gary Oldman, Dan Day-Lewis and Ken Branagh - do really well in film and be brilliant at it."

There were also movie directors he wanted to work with, not least the Coen brothers. In 2008, he auditioned for a part in the Coens' film A Serious Man but didn't get it. He was upset (although he now admits, "There was no way I look Jewish enough" for the role Michael Stuhlbarg eventually played). Soon after, he was in another terrible film calledBlitz.

Filming one scene in which he endured take after take of another character being sick in his face - "pea soup everywhere" - he decided he was finished with movies. He sacked his agent.

But since he stopped trying to sell himself to Hollywood, and particularly since his daughter died, the film business has approached him "in quite surprising ways". He believes these facts are interrelated. Nataasha was a film director. She always wanted him to do more movies. "A lot of the good fortune I'm experiencing now, it feels like I'm getting help," says Rylance. "And in fact, you know, to have Spielberg call me at home in Herne Hill to talk about his film. I can't tell you how many times I have joked when the phone has rung and actors have been, 'Oh, it's Spielberg, for you.' That's been a running gag in my life. Or for Sean Penn to call me up and say, 'I want you to do this film. Eight days. A ridiculous amount of money.

You know, we'll work around you.' These are jokes. These are Nataasha, in the universe, laughing her head off at me."

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Spielberg might make Rylance more famous than he is now, but it won't be the work for which he is principally remembered. Whatever choices Rylance makes, the greater part of his legacy will be in the theatre. It's shocking to learn that this man who seems so at ease on stage has endured several moments in his career when he has found it almost impossible to leave his dressing room - the actor's equivalent of the yips. One of those debilitating periods came during the second London run of his greatest triumph: when he was playing Rooster Byron in Jerusalem at the Apollo. He couldn't look his fellow actors in the face. A voice in his head told him to apologise to the audience for being such a terrible actor. He stopped taking curtain calls. "It was a kind of mental illness," he says. But the show, I say, had been such a success. What was he worried about? "I thought they'd all been lying. 'Why are they laughing? I'm going to kill every laugh in this thing. They're laughing because they've been told it's good. None of them are really laughing.' That's what I thought."

Mackenzie Crook, who shared a dressing room with Rylance for more than 400 performances of that run, remembers the period well. "Something was happening," remembers Crook. "Some crisis of confidence. And the way it manifested itself was that he withdrew into himself and he was very silent. I never felt it was directed at me or anyone else."

"He lost his temper on stage. He was on the brink of losing control." (Co-star Mackenzie Crook)

At times, though, the darkness threatened to overwhelm Rylance. "There was one particular performance which was a bit harrowing," says Crook. "That was the worst point of this crisis.

The anger bubbled out. He exploded and threw something and smashed the window on the caravan. He lost his temper. And that was kind of scary because he was on the brink of not being in control. But it was amazing theatre."

Even in happier times, Rylance's talent can cause problems. He is magnetic, and not every actor wants to spend his evening in another performer's orbit. Rylance tells a story about a touring production of Measure For Measure, in which he - as Duke Vincentio - had played with the audience by bumbling his lines for comic effect. At the closing-night party, Liam Brennan, who had played Angelo opposite Rylance's mumbling Duke, drank himself into a morose mood. When Rylance found him in the early hours of the morning, he was complaining, "Mark, why do you do it? You used to be my hero. Why do you do it?"

Mike Alfreds says that Olivier had the same effect. Even when he was playing small parts, "He could not but be interesting," and this created difficulties for the other actors. "Mark sincerely does believe in the company and he's got a great ethic in that sense," says Alfreds. "When he's in rehearsal with you he's so generous, he loses any sense of status. But his energy!

He's like a wonderful football player. He's just so much more talented than everyone else. He demands that you play up to him."

At The Globe, where Rylance was the first artistic director, and where he was a paternal figure as well as a colleague, he needed to give of himself in unusual ways. If there was an illness or an actor was away, Rylance would go on stage and play the part with a book in his hand. He remembers one occasion where he had performed as Cleopatra in the afternoon and then had to play two of the twins in The Comedy Of Errorsin the evening. Those nights were chaotic, improvisational and at times inspired. Indeed, Rylance's daughter Juliet maintains that Rylance's book-read performances are some of his best. "It's something to do with the comedy of them," says Rylance. "It was a breakthrough. It was like you were seeing what I do in rehearsals. You were seeing something rawer."

The Globe experience taught Rylance many things about where his strengths lay. His acting, already outstanding, became thrillingly free. He played with the audience more. Marshall Sella remembers that during the Broadway run of the farce Boeing-Boeing in 2008, Rylance's high-school ice-hockey coach came to see the play and the actor mimed skating across the stage as a wink to his old teacher.

More importantly, he realised he was not naturally suited to being an administrator. Especially, he didn't like dividing his duties between being a brother-actor and being the boss. He took to wearing a special "artistic director" hat (a trilby) to make the distinction clear to his colleagues. Nevertheless, there were times when the two roles became blurred. Sometimes, he would finish a matinée performance of Hamlet at 5pm, then chair an executive meeting at 5.15pm. "I'd still be sweating. I wouldn't have even had a shower. I'd be in a small room with people talking about catering or something... I couldn't explain to them that I'm going to be emotional, I'm going to be paranoid. I will just have died."

Rylance's combustible nature eventually became too much for The Globe to bear. The board knew when they hired Rylance that he was a sceptic on the question of whether "The Stratford Man" wrote the Shakespeare plays. Rylance says now that he simply keeps an "open mind", because it's helpful to view the plays as products of the ideas of an entire era, rather than just one man. (This is too big a subject for this piece, but for what it's worth, I don't buy Rylance's hedge. Most Shakespeare authorship-denial stems from a belief that an actor from Stratford could not have had the genius or the life experience to write the plays, and Rylance believes the problem with the Stratfordian view is that it "limits the plays to the mind of their creator".) In any event, his views on the issue became vexatious when he asked for books advocating different interpretations of the authorship question in The Globe shop.

Rylance was also vehemently against British involvement in Iraq and began to see the board members as emblematic of an Establishment that supported the war. He remembers one meeting in which a city banker laughed at him for saying he was a pacifist. "I thought, 'Are you not pacifists?' The ideal situation being that there isn't war, that we're not killing each other?' I mean, if someone attacked my wife and kids, and I had a weapon, I might well kill them. I hope I wouldn't. But I understand that our nature is violent. But surely the ideal is that we sit down at the table and talk sooner rather than later?"

When Rylance left The Globe in 2005 - "very depressed, very tired" - what he had built there could not be destroyed. It had become a venue for Shakespeare with a broad appeal, that re-created to a large degree the experience of an Elizabethan audience. It was a place where you could pay £5, stand in the yard, and watch world-class actors play Shakespeare using "original practices": not only acting on the same kind of stage, but wearing the same kind of clothes, using the same kind of props and playing the same kind of music. For Rylance, the building was like a second home. His daughters had spent so many evenings there, it was as if the circle on the South Bank was a kind of surrogate parent to them. He could not have imagined that it would take him seven years to perform again on its stage, or the circumstances under which he would do so.

The summer of 2012 had been hard for Rylance, but the plays at The Globe were a hit, and runs in the West End and then Broadway were booked. In February 2014, on the final night of playing the grieving Olivia in Twelfth Night in New York, while his family sat in the audience, something strange happened to Rylance. He remembers that he became aware of another voice in his head - that of his mourned-for daughter Nataasha. "I found myself doing funny things, like tilting my head and finding myself being amused by funny things," he says. "And I suddenly realised, these are things that Tash used to do. This is Tash's way of being amused. And I had a strong sensation suddenly that she was inside me. She didn't have to sit in the audience with the rest of the family. She was able to come inside me and ride along through the play with me and give me little ideas.

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"I found myself looking at other actors with Tash's voice in my head going, 'Oh, that's very funny. Oh, they're acting. What are they doing? Oh, this is very curious, Pops,' and riding along with me in the play, rather than being at a mournful distance. That actually she was right here with me. Not far away. In many ways closer than the members of my family who were in the audience. That she was liberated from the separation of physical existence and was fluidly living within me now... So the end of that two-year process of doing those Shakespeares gave me a great revelation about where my youngest daughter is now, for me. It's not to say that I don't miss her terribly. I do. But I also have experiences of her being very close, like that."

Rylance told that story in a high voice, like the one he used as Olivia in Twelfth Night. His eyes became glassy. He was reliving the moment, becoming lost in its spectral wonder. The soliloquy gave me gooseflesh. It didn't matter that in my everyday life I gave no more thought to the spirit world than I did to the existence of aliens. When Mark Rylance told me that he was possessed by his daughter while on-stage at the Belasco Theater at West 44th Street, New York, he believed every word, and so did I.

Mark Rylance appears as Thomas Cromwell in Wolf Hall airing on BBC Two, Wednesday 21 January