Beyond the péripherique, a short drive through the Bois de Vincennes – by Paris's close-cropped standards almost unimaginably untamed – is the Théâtre du Soleil, the company Ariane Mnouchkine co-founded and has run for the last half-century. The theatre is located somewhere in the grounds of the Cartoucherie, a former munitions factory buried in the dark of the wood.

For many, the journey is a pilgrimage. Mnouchkine is regularly ranked as one of the world's most influential directors, the only female winner of the international Ibsen award, whose productions – 20-odd in the last 48 years – are often spoken of in awe. Soon after the événements of 1968, Théâtre du Soleil exploded to attention with its rough-and-tumble devised works 1789 and 1793, which brought the history of the French revolution to scandalous, shocking life. Their cycles of Richard II, Twelfth Night and Henry IV (1981–84) reached an estimated 250,000 people, fusing Shakespearean history with stylised imagery drawn from Japan, Bali, India, then a four-part odyssey into Greek tragedy, Les Atrides (1990–92), that blended Kathakali with kabuki.

Always restless, Mnouchkine has sometimes made migration itself her subject: 2003's Le Dérnier Caravansérail (The Last Caravan Stop) dramatised the experience of refugees from Kurdistan, Chechnya, Iran, Russia and a flurry of other countries into an extraordinary six-hour experience that featured no fewer than 169 characters. In French, the word spectacle is simply the offhand word for "show", yet Théâtre du Soleil interpret it literally – they seem unable to make theatre that is anything less than epic.

But for all its flamboyant internationalism – 21 nationalities are represented among Théâtre du Soleil's ranks, and it now has sister companies drawn from Afghanistan and Cambodia – the company has only ever visited Britain twice, creating the intriguing paradox that more people here have read about their work than can ever have seen it (not least because Mnouchkine almost never grants interviews to the English-speaking press).

When they bring their sea-soaked Les Naufragés du Fol Espoir (Aurores) to the Edinburgh festival later this month – teasingly described as "a play collectively devised by Théâtre du Soleil, half-written by Hélène Cixous [and] loosely based on a mysterious, posthumous novel by Jules Verne" – it will be the first time they have toured to the UK for 20 years. It is quite a moment. The Ibsen citation declared: "to enter a Mnouchkine production is to enter another world." In the ringing phrase of the critic Judith G Miller, she is nothing less than a "nomad of the imagination".

If one were looking to cast a nomad of the imagination, the mild-looking woman in front of me, swathed in a baggy grey hoodie, a tussle of white hair coiling around her spectacles, isn't an obvious candidate for the role. It has been arranged that we will talk after the company has lunched together. Though the Cartoucherie is not the commune it once was, this isn't so much a meal, I sense, as a metaphor for how the Théâtre du Soleil operates. "Well, it's important," Mnouchkine replies in softly accented English. "We work a lot, so life has to be as nice as possible, and relationships too – we are a company of 75 people now, and of course there are crises, problems." So it's not entirely the workers' paradise it seems? She laughs. "Sometimes it's hell! Sometimes it's terrible. It is made by human beings. They're not angels, and they're not devils either. They can be both. We can all be both."

It's tempting to speculate that this urge to forge a family – and her lifelong fascination with migration – has its roots in Mnouchkine's own past. She was born on 3 March 1939, to a British mother, the actor Jane Hannen, and the film producer Alexandre Mnouchkine, a Jewish–Russian émigré who fled to Paris with his parents after the revolution. Mnouchkine rarely talks about her mother (her parents separated when she was 13) and perhaps the most important relationship in her life was with her father, who became one of France's most successful postwar film producers, and remained until his death in 1993 an intimate artistic collaborator, helping finance her first theatre, scraping together cash for what remains her only full-length feature, 1976's Molière. "At every moment of my life," she said in 2005, "the only thing of which I was entirely certain was that my father loved me."

After France crumbled to the Germans in June 1940, the young family was forced into hiding; Alexandre's parents, unable to escape, were dragged from their home, then deported and gassed. Mnouchkine's only authored script is an adaptation of Mephisto, Klaus Mann's bitterly satirical novel about the rise of Nazism, and her deep sympathy for displaced peoples and the victims of conflict is not confined to the stage. She has been a passionate campaigner for human rights, especially in the Palestinian Territories, Tibet and Afghanistan, and in 1996 the company opened its doors to 382 illegal immigrants, many from the former French colony of Mali.

Though the young Ariane grew up watching Parisian theatre – it was there that she first saw Kathakali – it's surprising to hear that she got a taste for performance in Britain (though there is admittedly the family connection: the actor Harriet Walter is a cousin). While on a year abroad from the Sorbonne in the late 1950s, studying English in Oxford, she drifted into the ambit of the Oxford University Dramatic Society. She recalls: "I was getting a bit bored because I didn't have enough to do, so I joined. I worked with Ken [Loach] and John [McGrath] as an assistant, we became good friends. Anthony Page came to direct Coriolanus." What was it that caught her about theatre? She sounds almost dreamy. "It was a realm, an island – a world that could be transformed, clarified, harmonised, analysed. I still remember the night I decided that theatre was going to be my life."

Back in Paris in 1959, and still a student, she set up ATEP (l'Association Théâtrale des Etudiants Parisiens) – a leftist rival to the Sorbonne's classical company that promptly made itself notorious by sponsoring political lectures during the Algerian war. It was this group that evolved into the Théâtre du Soleil, founded as a cooperative in the summer of 1964, with each member contributing 900 Francs (salaries are still more or less equal across the company, and far from generous: €1,400 a month for new arrivals, €1,800 for long-term members, including Mnouchkine herself). Naming the theatre after the sun, Mnouchkine told the New York Times in 1992, felt inevitable: "We were looking for life, light, heat, beauty, strength, fertility."

A scene from Le Dérnier Caravansérail at the Theatre du Soleil.

It was a British play that gave them their first hit – a production of Arnold Wesker's The Kitchen, performed in 1967 in the radical setting of the Cirque Médrano, whose unsparingly, precise choreography, learned from a stint with the circus guru Jacques Lecoq, introduced smart Parisian audiences to the unvarnished realism of this angry young Englishman (albeit one who had sweated in the real-life kitchens of Paris's Restaurant le Rallye). "I'm still so grateful to Arnold for giving us this play," Mnouchkine smiles. "It was a big success in England, we were really nobody, and he gave us the rights. Suddenly we existed."

Nonetheless, it was a canny bit of planning. Wesker's script makes pointed links between the broiling tensions in the workplace and those of wider society; by May the following year French students were on the streets, and TdS threw itself into la lutte, performing the play for striking car workers. But though the company had found a purpose, it still didn't have a home: a struggle to take over the defunct Les Halles market ended in failure (Mnouchkine still grimaces at the memory), but in 1970 they got a tip-off about the old armaments factory in Vincennes: "It was a miracle, really. Somebody told us about it, and I arrived here the same day. A soldier was cleaning up and I said, 'What are you doing?' He said, 'I'm closing up, the army has gone, I'm the last one.'" She looks triumphant. "We squatted before he left."

The company set about the arduous process of transforming the derelict units into living, breathing spaces: accommodation for anyone who needed it, prop manufacture and storage, wardrobe – and of course, the theatre itself, a huge hall under an expansive glazed roof that is internally rebuilt for every production (for Edinburgh audiences a life-size replica is currently being raised in an exhibition hall near the airport).

Later, I am escorted by an actor through the Cartoucherie's treasury of converted warehouses: a workshop for musician Jean-Jacques Lemêtre, stuffed with improbable-looking instruments; cavernous costume stores, with shoes and camisoles neatly filed in boxes on one wall, fur and velvet capes beneath blankets on racks; a mask workshop.

People who buy a ticket to the Théâtre du Soleil don't simply get a show: they are absorbed into a dramatic experience whose every detail, from the navette that picks them up from the Métro to the meal they eat beforehand, has been carefully, almost ritualistically planned. Members of the company cook and serve; visitors are invited to peer through windows into dressing rooms to observe the cast putting on their makeup. This is one of the few theatre companies where audiences are required to put in nearly as much mental preparation as the actors; one of the earliest, too, to explore the concept of theatre as immersion, decades before Punchdrunk and other such companies.

Mnouchkine explains: "We're not a shop. It's unfortunate we need people's money, but we're not selling something. That's why I hate the word 'production': it's not produced, it's a ceremony, it's a ritual, it is something which is very important for your mental strength, and you should go out of the theatre stronger and more human than when you went in." And she herself still doles out food and tears tickets? She pulls a face. "The least you can do is welcome people."

Although Mnouchkine's career is sometimes separated into phases – from the early, improvisatory works, through the epic sequences of Asian-influenced histories, to her border-crossing collaborations with writer and philosopher Cixous, who first became involved with the company in the mid-80s – the company's lifeblood has been improvisation. Nearly everything you see on stage is created collaboratively – in the early years laboriously observed and notated from rehearsals, these days captured by a small video camera. She looks scornful when I ask why improvisation is so important. "It is a strange question, because would you ask an author why ink is important? It's a way of writing. In our company, actors are really the authors." Even when they're dealing with a pre-existing text? "Theatre is not only what you say, it's what you do. It's the reaction that has to be caught, it's the mood, the emotion, the state of mind. Even when we worked on Shakespeare, we improvised. Otherwise you just repeat famous phrases. What you have to do is try and understand not only how Shakespeare wrote it, but how he heard it." She mimes a splat on the table. "What did he feel when he suddenly discovered, say, Falstaff landing on his paper?"

Mnouchkine experienced her own thrill of discovery when she first travelled east, taking a year off in the early 1960s to explore Japan, India, south-east Asia and, bravely for a 23-year-old travelling alone, remoter parts of the Middle East (she remembers seeing women accessorising their hijabs with miniskirts in a pre-Russian Afghanistan). It opened the young director's eyes to art forms as richly diverse as bunraku puppetry and Javanese gamelan. "It was a shock," she reflects. "It was like suddenly finding what you knew existed but hadn't seen. Asian theatre is a master for metaphor."

It was years before those metaphors crept into her own work, initially in the Shakespeare productions, but the lessons were profound. "It took some time to digest everything, to make it ours, not what we call Japonaiserie." There was, however, criticism that this was exactly what the productions were – well-meaning but naive attempts to borrow ideas from the east and attach them to western forms. "In Richard II there was still a little bit of Japonaiserie," she admits. "We use what we need. We're like a bunch of mécaniciens foux, mad mechanics, who pick up this tool, then that other one, use it in strange places." She remarked once, I say, that Asian theatre helped her escape the trap of naturalism. She smiles cryptically. "Naturalism is a good word for a bad idea. Art is to do with transformation."

Like a surprising number of directors, Mnouchkine seems touchingly in awe of the powers exhibited by actors. I wonder if she ever feels envious. "When I'm on stage, I don't believe myself," she says. "But when I look at them I believe. Even someone who is supposed to be a king but looks like a potato."

She resists talking about her family, her father aside; although she had a relationship with Cixous and, after they went their separate ways, another female member of the company, her private life remains private. While almost childlike in her warmth, there is a trace of remoteness about her. When she won the Ibsen award, she shared the €300,000 prize money with her colleagues, so they could have an extra month of rehearsal. I wonder where the company ends and she begins. Perhaps it isn't a distinction that means much.

One answer might lie in Les Ephémères (2006), roughly translatable as "Ephemera", described by one critic as a show in which "the grey fabric of everyday life … is made to shimmer with significance and quiet intensity". In some ways her most intimate piece to date, it was a mosaic of narrative fragments played out – rhythmically, heartstoppingly – on wheeled stages spun by hand through the darkened space: an image of how individual memories become history, and also of their elusiveness. Several scenes emerged from memories Mnouchkine offered herself: one, perhaps rawest, of her mother saying goodbye to their family. She admits she was taken aback to see it creeping into the show. "It was by accident: all the actors were telling their own memories, so I found myself obliged, in a way. And then one actor suddenly grabbed it and put it on the stage." Was it strange, seeing her own life represented after a lifetime representing others? "Yes, but also rather beautiful. The person who was really surprised and moved was my sister. She had no idea: I was so occupied with everything that I didn't think to warn her."

Mnouchkine has fought many battles during her 73 years: getting the company going, finding it a home, travelling, researching, giving political press conferences, housing refugees; once, in 1995, taking part in a month-long hunger strike to protest France's failure to intervene in Bosnia. She remarked in an interview a few years back that she felt that her "forces of consolation [were] diminishing". What did she mean? "It's true," she sighs. "I don't feel my age except in some parts of my body, and in that it takes more time to se conseler, how do you say, console oneself? When I was young I was violent, in a way. I would break sorrow." She mimes boxing. "I don't do that any more, I just put it away. Do you understand what I mean? It takes a lot of strength to get over sorrow." Her voice softens. "I still do it, it just takes more time."

Her theatre has sometimes been called utopian. Is that a good thing? "Utopia is not something which is impossible; it is something that has not been done yet." So somewhere utopia exists? "Yes," she says, with disarming definiteness. "We are on the path to utopia. I accept that."

• Les Naufragés du Fol Espoir (Aurores) is at the Edinburgh festival from 23-28 August. www.eif.co.uk/folespoir.