When Tony Blair relaunched himself in 2016 – out went the consultancy for questionable oligarchs, in came a focus on big, vague Miss World-like topics such as “governance” and “peace” – his Institute for Global Change chose a provocative slogan. According to its website, it wants to make globalisation “work for the many, not the few”. Remind you of anyone? Yes, it’s the tag line used by Jeremy Corbyn during this year’s election – but it was taken from New Labour. Now, it seems, Tony Blair wants his soundbite back.

We meet at the institute’s offices in Bloomsbury, central London. There are acres of cream carpet, tasteful Middle East cityscapes, delicate Japanese screens and abstract sculptures that often reveal themselves, on closer inspection, to be awards from worthy international groups. The message is clear: Labour’s greatest election-winner might be a pariah in Britain – a recent appearance on The Andrew Marr Show was not advertised the night before, as guests usually are, because of security concerns – but his expertise is still valued abroad.

We talk about foreign policy – China, Iran, America – but it’s clear that Brexit and Labour are irresistible subjects. The great persuader still wants to win us over; perhaps more so since “Blairite” became a stronger term of abuse on the left than “Tory”. The 2017 model Blair is particularly interested in technology and the rapid pace of change, and his institute has just published a report recommending better regulation for social media, lifelong education for anyone whose job is lost to automation and a government department for digital matters. (Quite a departure for a man “named and shamed” in 2003 by the Conservatives as the only world leader without an email address.)

Personally, he remains agnostic about social media, which has helped “develop the era of the loudmouth”. While he praises Jeremy Corbyn’s success on Facebook and Twitter for diminishing the power of the right-wing press – no need for yacht trips with Rupert Murdoch for Jez We Can – he is quick to add: “Exchanging tweets is not the way to debate serious policy. It isn’t.”

A former Middle East envoy for the Western powers, Blair diverges from the consensus view on Israel-Palestine, saying that he is cautiously optimistic about the two-state solution. “The Israelis and the Arabs actually have a huge strategic interest in common, and really both have the same essential obstacle to progress, which is what you might call Islamist ideology,” he adds. “So you’ve got two things happening: one is an anxiety that they share about Shia-inspired extremism on the one hand, through Iran, and Sunni extremism emanating from the Muslim Brotherhood through to Isis.”

He sees the rise of Saudi Arabia’s 32-year-old crown prince, Mohammed Bin Salman, as “the single most important thing that’s happened in the last few years in the Middle East”, describing him as a “reforming moderniser”. However, the prince has not yet dialled down the broader Sunni-Shia conflict, with many seeing Saudi Arabia’s attacks on Yemen as a proxy war with the Shia regime in Tehran.

Iran “is a problem”, concedes Blair. Now a private citizen, he visits the Middle East twice a month and says that the key to understanding its problems is “to understand that it’s essentially one struggle. It’s about whether these societies that are full of religious tribal tensions with poor systems of government and institutions can move towards religiously tolerant societies with rule-based economies.”

How much, though, should other countries involve themselves in that process? I tell Blair I find it strange that David Cameron gets so little criticism for Britain’s 2011 intervention in Libya, which removed a dictator, created a power vacuum and led to widespread violence and lawlessness. Wasn’t removing Muammar Gaddafi repeating the mistake we made in Iraq? And if so, why does one create so much more anger than the other? “The simple answer on Libya is that we didn’t put troops on the ground,” he says, flatly.

I suggest that Britain’s current isolationism offers an easier moral accounting: we do not feel responsible for violence, sectarianism and famine in the countries in which we did not intervene, as if inaction were a neutral choice. “Absolutely,” he replies. “What we’ve allowed to happen in Syria is absolutely terrible… I mean, what’s extra­ordinary to me is that the destabilising influence of Iran around the region is colossal, and yet… progressives have sort of lined up, essentially, in a position of saying, ‘Look, you’ve just got to step back from the region,’ or saying, ‘Don’t interfere in Syria,’ when the interference from the other side in Syria is vast.”

The “other side” here means Russia, which has consistently backed the brutal dictator Bashar al-Assad. (Blair will not be drawn on whether he thinks Vladimir Putin also meddled with pro-Brexit propaganda, saying: “I don’t know, but I think we should find out.”) Yet the left’s uneasiness with intervention is surely not just a result of post-Iraq regret, but also a rejection of lining up unquestioningly alongside America, as if the old Cold War axis still holds.

Blair believed in the special relationship – and had good relations with Bill Clinton and George W Bush – and is hard-headed about keeping the faith while Donald Trump is in the White House. “The American president is the American president, so you can take whatever view you want to take, but in the end you’re going to have to have a relationship with him if you’re the British prime minister. That’s why I don’t attack Theresa May for having a relationship with Donald Trump. She’s got to.”

This articulation of realpolitik is now deeply unfashionable on the left, as is the assumption that a US-UK alliance can offer moral guidance to the rest of the world. Is the dominant strain in left-wing thought now anti-American and anti-imperialist? “Yes, it’s anti-West. It’s not anti-war, otherwise they’d be demonstrating out in the street against Assad, and they’re not.”

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Tony Blair and Jeremy Corbyn entered parliament in the same year, 1983, and their utterances from that period do little to distinguish them. Blair even paid homage to Euroscepticism, then the prevailing mood in Labour, writing in his 1983 election literature that the EEC had “drained our natural resources and destroyed our jobs”. It was Blair rather than Corbyn who wrote to Michael Foot in 1982 to say that he “came to socialism through Marxism” and that he agreed with Tony Benn that: “The right wing of the party is politically bankrupt.”

Over the ensuing decades, Corbyn opposed the Maastricht and Lisbon treaties, while Blair became steadily more open about his own Europhilia. The New Labour leader became a devotee of “liberal humanitarian intervention”; the rebellious backbencher became the chair of Stop the War.

What Blair and Corbyn share is the ability to make a deeply unpopular case in the teeth of great opposition. In his 2010 autobiography, A Journey, Blair writes: “Labour Party politics following the defeat of 1979 was a bit like revolutionary France at the time of the Thermidorian Reaction, full of infighting, intrigue and bitter recrimination. The MPs were regarded by a large section of the party as sell-out merchants who had ‘betrayed socialism’.”

In Blair’s account, New Labour triumphed because he and his fellow MPs learned to argue their cause – and his inspiration was Corbyn’s early mentor Tony Benn. Blair writes of watching Benn speak in 1983 and being captivated by his confidence, sense of humour and the “thread than ran throughout the speech… The argument was built, not plonked down.” Still, the younger man decided at the end of the night that Benn was missing one thing: “He was the preacher, not the general. And battles aren’t won by preachers.”

As the Brexit process grinds on, many on the Remain side would be grateful for a general. But for the moment they must settle for preachers – the former Lib Dem leader Nick Clegg, a smattering of liberal Tories and Labour backbenchers, and Blair himself, who seems re-energised by the cause.

He points to the “humiliating” fact that Britain was unable to get its chosen judge nominated for the International Court of Justice at the UN in November. (Christopher Greenwood’s failure in a run-off against the Indian nominee means that it will be the first time since the court was founded in 1946 that there is no British presence there.) The relocation of the European Medicines Agency – along with 900 jobs – from Canary Wharf to Amsterdam “makes me weep”, he says. “What’s happening is the most irresponsible abdication of leadership I have ever seen in my political lifetime. It’s just incredible.”

For Blair, Britain’s new isolationism shows our politicians’ inability to defend our alliances and to make the case that we are stronger when we co-operate than when we compete. “There is an insanity about this on the trade side,” he says. “I mean, the idea that Britain out of Europe is going to conclude a better trade deal with America, with China, with India…”

At a time of increasing protectionism, harder borders and anxiety over immigration, the mission statement of the Institute for Global Change is to defend the concept of a connected world. “The thing that depresses me about the state of the politics here today is our leadership isn’t explaining to people, ‘This is the way the world is changing,’” he says. “It’s like if you said to the owners of Manchester United and Manchester City, ‘From now on, you can only take your players from the Manchester metropolitan area,’ and they say to you, ‘Well, I’m afraid in that case, we’re not going to be winning the Premiership,’ and you say, ‘What, you’ve got no faith in Mancunians? You don’t believe in them any more?’ I mean, this is to confuse patriotism with a sort of delusion about the way the world’s changing.”

Brexit is a symptom of this disease. “We probably will get a deal of some sort,” he says. “But it’s going to be ugly and damaging.” He sketches out the six possible endgames: first, admit we made a mistake and stay in the EU. Second, negotiate reforms and stay in the resulting Europe. (“This would be my preferred option.”) Third, leave the political institution but stay in the single market and customs union: as a rule-taker but not a rule-maker.

“The fourth option is really where Theresa May and Philip Hammond and people are, which is that they want to leave Europe but still retain close ties to Europe,” he adds. “It’s a kind of ‘leave without leaving’ strategy. I personally think that’s incredibly difficult to negotiate, far more than they think at the moment.” The fifth option is to leave and “market yourself as non-Europe” – the course threatened by Hammond earlier this year and beloved by the libertarian wing of the Brexit movement. The sixth option is to leave with no deal at all.

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Both Labour and the Conservatives appear to have ruled out the first three options. “And that, to me, is just another abdication of leadership,” Blair says. May triggered Article 50 “literally without knowing what we wanted”, so the fight now is to expand the options – “Otherwise, you’re left with four, five or six, and four is very difficult.” On the single market, Labour has a position very close to that of the Conservatives, maintaining that “access to” it is a viable substitute for membership. “Yeah,” says Blair. “I hope that the leader’s position is a tactic and not a strategy. We’ll see.”

Another open question is whether May is strong enough for her premiership to survive the negotiations, and then to get her deal through parliament. Blair believes her administration “is in more disarray and is more hopeless than any government I’ve ever seen”. Even the last days of the John Major era? “It’s sort of worse than that. Ken Clarke was chancellor. He was actually taking some reasonably sensible decisions… If you actually reflect on it, to have a prime minister and a chancellor driving through a project that they fundamentally believe is a mistake is a pretty weird mission.”

The difference between the mid-1990s and now, of course, is that back then Labour moved solidly ahead in the polls. At the election in June, Corbyn’s Labour was able to pick up the votes of frustrated Remainer swing voters, while holding on to its Leave-leaning northern English bedrock. Electoral logic therefore dictates that it’s better for Labour to appeal to skittish Leavers than to hard-core Remainers, since the collapse of the Lib Dems has left the latter group with nowhere else to go.

Blair dismisses this, arguing: “If Labour were really making an issue of Brexit in the right way… you could then lead the people who were Labour people that voted for Brexit to an understanding that Brexit’s not the answer to their problems.”

Underlying this is an acknowledgement that these are hard times for any party with a redistributive social programme. The November Budget included growth forecasts of under 2 per cent for the next two years; social mobility has stalled; real wages have not risen in a decade. No wonder there is a hunger out there for something more revolutionary than vanilla social democracy.

“The Labour Party…” Blair briefly pauses. “There is a void at the heart of its argument at the moment, which is the view that, ‘OK, we’ll do Brexit, but then we’re going to have this great radical programme,’ but actually the two are in conflict. It’s a very simple thing. For the health service, for example, in many ways you’ve got to choose between: do you want to be rebuilding the health service, or do you want to do Brexit?”

On social mobility, “The answers are not to get rid of immigrants, right?” – but to improve education and infrastructure. He says: “I don’t see in either party at the moment policies which would radically address those two issues.” Along with many in the Parliamentary Labour Party (PLP), Blair is sceptical that abolishing tuition fees is the best use of £8bn, when it is non-graduates whose wages have been most remorselessly squeezed in the past decade. Unlike his ideological sympathisers in the PLP, however, he is willing to say that out loud: “If you want to spend [£8bn] on education, you should surely be concentrating on early years.”

Hardened by the battles of 1980s, Blair is less inhibited about challenging the left’s ideas than his intellectual heirs have been. (There are many older Labourites who lament that the years of New Labour hegemony created a soft generation who never learned to argue their corner.) Corbyn, for his part, has never stinted on criticisms of Blair – but has always stayed in Labour.

I end by telling Blair that I watched the 1997 election broadcast this year, on its 20th anniversary, and was struck by how happy Jim Callaghan and Neil Kinnock – who came from different Labour traditions – were to see a Labour prime minister going into Downing Street. Will he be happy to watch Jeremy Corbyn walk through that famous black door?

For the first time, the smooth delivery falters. The “look” count spikes. “Look, I don’t hide my disagreement, because I have a disagreement, and my disagreement is over what progressive politics should mean in the 21st century,” he says. “I struggle to see this as a project that will work in progressive terms… You know, but I may be wrong. Let’s see.”

But it would still be better than a Conservative government, right? “Look, I’ve always voted Labour and will always vote Labour, but I worry about aspects of the policy.” He thinks the mistake that Corbyn’s opponents made was to dodge the argument about whether the leader’s ideas were right by instead claiming that he was unelectable. (If so, it was a mistake that Blair made himself: during the leadership contest in 2015, he warned that Labour faced “annihilation” under Corbyn. Instead, it gained 30 seats this summer.) That acknowledgement defines his mission: researching and advocating a radical set of principles distinct from what he sees as Tory Brexitmania and Labour’s statism.

Warming to the theme, he adds: “It’s like when I hear people say the last Labour government was a neoliberal government. I mean, it’s ridiculous.” (His breezy confidence when making this point suggests that he is telling the truth when he claims not to look at Twitter.) “The minimum wage, massive investment in health and education. When we left office, levels of NHS satisfaction were the highest they’d been since the NHS began, apart from a brief period after it began.”

In a recent podcast interview with the former Obama adviser David Axelrod, Blair made a striking claim: if Clement Attlee woke up in modern Britain, he would be amazed by the change he would see everywhere – except, that is, when he walked into Whitehall. “Exactly,” he tells me. “Government should be completely re-engineered for today’s world, and it should be strategic and empowering, not big and controlling. And this is a progressive argument.”

Does anyone want to hear it – and from him of all people? “If you look back at that Labour government, OK, if you want to, just focus on Iraq… but there were masses of progressive things that were done.” Our time is up, and I pack up as he lists them: gay rights, constitutional reform, changing the House of Lords. “But we only managed to do them because we won,” he says. “And we won successively.”