It wasn't quite the image of chanting crowds, banners and social media-savvy youth associated with the anti-establishment left. But when Professor Richard Tuck, a Eurosceptic Harvard academic, came to speak before the Policy Exchange think tank last month, it was a radical vision on display.

He spoke of the left having its “greatest prize in a couple of generations, with the possibility of genuinely transforming British politics” through Brexit. Tuck was only the latest of a number of pro-Lexit – left-wing Brexit – intellectuals doing the rounds with their visions.

For left-wing Leavers, improbable victories have been coming thick and fast. With first the EU referendum, and then Labour’s impressive performance in the general election, led by Jeremy Corbyn, a man perceived as a closet Eurosceptic even during the Remain campaign, pro-Lexit views are being taken with newfound seriousness.

“Brexit means Brexit,” as Theresa May so famously said. It’s looking increasingly possible that Brexit could also mean Lexit. But what would Lexit look like?

During the campaign, Labour Leave – an organisation of Eurosceptic Labour MPs as well as donors like John Mills – released Lexit: the Movie, which framed the campaign as one of “the market versus society”. Labour Leave became active after Mills and others who had been involved with the wider campaigns Vote Leave, Leave.EU and Grassroots Out decided, in Mills’s words “Labour voters needed a Eurosceptic home.”

Yet others on the Left – including historically Eurosceptic parties such as the Communist Party of Britain, and the Socialist Workers Party – also claimed the “Lexit” mantle. They formed their own “Left Leave” campaign, which would be backed by some Labour-affiliated trade unions who opposed Labour Leave sharing platforms with Ukip.

The chair of Left Leave, Robert Griffiths, says his organisation received “plenty of invitations from trade unions and trade union councils” to speak and debate against pro-Remain campaigners. But it also organised a convoy to Calais to combat what he calls the “anti-foreigner kind of nonsense.”

Underlying these divisions is a split between those who see the sovereignty of the UK’s “national democracy” as an end in itself, and those who view it as a means to an end of carrying out left-wing policies.

Many Labour Leave campaigners fit into the first bracket. In the Labour MP Graham Stringer’s words: “I'm pro-democracy based on sovereign states, not on some super authority over and above democratic governments making laws in an unaccountable way.”

When asked for his view of an ideal Brexit, Mills said it was “pretty well in line with the speech that was made by the Prime Minister at Lancaster House”, in which May confirmed the UK would leave the single market, control immigration but seek new free trade deals. “Lexit” may not be a useful term to describe a Conservative leader’s line of opinion – but Mills indeed argues that Brexit is distinct from left-right politics. “I think to a large extent it is a distinct issue,” he says. “I don’t think there’s much of a redistributive content for example to do with Brexit.” Instead it is about “what extent you think you ought to be part of the European Union rather than an independent country”.

Yet for others, Brexit is a means to a very ideological and redistributive end. Labour Leave backer Kelvin Hopkins says Brexit is a “necessary condition for a left government to be able to do what I think a left government should do”, which is to restore something like the postwar Keynesian system, which he calls “a world that worked”.

Immigration is another dividing line. Frank Field, a Labour MP who backed Leave in part to increase control of immigration, compares calls to guarantee the rights of EU nationals in the UK to poor wage negotiations in an industrial dispute. Voters will see this, he argues, as once again putting “Britain second to other needs”.

In contrast Ian Hodson, president of the the Leave-supporting Bakers, Food and Allied Workers Union, declares “one of the things about the Bakers' Union is that we actually don't believe there should be any borders anywhere”.

What does unify most of the Lexiteers is concern that Labour could adopt a position on Brexit that pleases nobody. Most, however, feel that their own influence over policy is slim.

Mills, the donor, says that “since the referendum, Labour Leave has had kind of a monitoring role” in order to “make sure that the decision taken by the outcome of the referendum is reflected in what Labour does in parliament”. Yet the MP Stringer admits that Labour Leave as a group has had “one meeting, a very informal meeting” since the referendum, in order to discuss their stance on the original Great Repeal Bill.

Few Leave supporters are in policymaking roles – one exception is John Hilary, Labour’s head of trade policy. When asked about his own influence over Brexit policy, Field replies: “None at all, sadly. I haven’t got any influence at all.”

For some, the Labour gains in the recent election are bittersweet, because it makes the chance of the government playing hardball in the negotiations less likely.

Yet while Remainers, who may be seeking damage limitation, comprise a significant part of the parliamentary Labour Party, many of the Lexiteers are convinced that the leadership is on side.

If Corbyn does push for a “Lexit”, it will likely be as a means to a left-wing redistributive agenda, and the general election has undoubtedly strengthened his hand.

As Hopkins, an old ally of Corbyn, puts it, in many recent elections “the similarities between parties are so great that people just stop voting”. In June, “this time people thought, well, whatever you say about Jeremy, he’s not like the Tories”. Now the Lexiteers must hope this extends to Brexit.