What’s the difference between Portsmouth and Liverpool? That’s not the start of the joke but the question that could yet see Britain crash out of Europe. Both are port cities where median income is below the national average – but Portsmouth, in the south, elects Conservative MPs by narrow majorities, while Liverpool, in the north, sends Labour politicians to Westminster by landslides.

As the local elections showed, elections are increasingly a case of “geography, stupid”. On 5 May, Jeremy Corbyn’s Labour party secured thumping victories in mayoral elections in Bristol, Salford, Liverpool and London. But outside England’s great cities, Labour stagnated or slipped back. Put crudely, the party does well wherever people put wind chimes in their doorways – and badly in places where WHSmith is the only bookshop.

At a general election, the distorting effect of first-past-the-post gives David Cameron the edge – but as far as the referendum is concerned, where all votes are worth the same, the Tories’ urban discomfort is the cause of considerable worry. As a recent study by the London School of Economics found, support for remaining in the European Union is strongest in Brighton, Bristol, Edinburgh’s Morningside district, Oxford and Hackney – all areas that rejected the Conservatives by heavy margins in both 2015 and 2016. Cameron can’t win over those voters to Remain – but Jeremy Corbyn could.

That is why Vote Leave, the official campaign to get Britain out of Europe, is planning to focus relentlessly on the issues that most concern swing Labour voters: reducing immigration and protecting the National Health Service. (It is no coincidence that these formed the centrepiece of Ed Miliband’s offer to the country last May.)

Strategists at Vote Leave believe that if turnout is low, they can bank on close to three million votes from Ukip supporters, add another three million from the Conservative column, and win over enough Labour voters to get them over the line. As one pro-European frontbencher put it, “we know what our problem is: a Labour-voting man at or over retirement age, who lives in the north”. It is men, in particular, who are keeping staffers at the In campaign headquarters at Dowgate Hill in the City of London up at night.

The Remain campaign is deploying a strategy to win over older women cribbed from Ireland’s same-sex marriage referendum. In “Talk to Gran”, young people – who back a Remain vote by large margins – are urged to write to and email their grandmothers encouraging them to stay in the European Union. (Why only grandmothers? Because research shows this approach doesn’t work on elderly men.)

Stronger In also has a strategy to drive up turnout among students, which is why Labour’s most committed pro-Europeans, and the Green Party’s only MP, Caroline Lucas, are spending increasing amounts of time on campuses. The campaign is also spending large amounts of money on Facebook advertising. The same approach, and the same personnel – Craig Elder and Tom Edmonds – helped to deliver victory for the Conservatives in marginal seats last May.

Yet among men, and Labour-voting men in particular, a strong statement from Corbyn could be decisive. Although Labour is officially committed to staying in the European Union, the strength of that institutional support varies from region to region. The Labour Party is largely pro-European, but only in the sense that the United Kingdom is largely Christian – only a minority believes with any ferocity.

In Wales, the party’s leader, Carwyn Jones, forbade candidates and activists from discussing the referendum on the doorstep during the assembly elections in an effort to blunt the Ukip surge. In the end, Labour retained all but one of its seats on the assembly and suffered no direct losses to Ukip – but the pro-European campaign in Wales now has weeks of work to make up.

That leaves Britain’s relationship with the European Union, in effect, in the hands of one man: Jeremy Corbyn. On paper, he could have been scripted by Stronger In as their emissary to Labour voters: visually and ideologically a very different type of politician from Cameron, strongest in the places where the Prime Minister is weakest. Even his long-standing Euroscepticism – as recently as June 2015, Corbyn told the New Statesman he had not “closed [his] mind” to a Leave vote – gives him a credibility in talking up staying in the European Union despite its flaws.

But that also means that Corbyn is well placed to usher Britain out of the European Union, simply by sitting on his hands and making a few grudging appearances at set-piece events, such as the launch of Labour’s pro-European campaign on 10 May.

Corbyn’s conversion to staying in Europe is one of Westminster’s most reliable Rorschach tests: people see what they want to see. Those of his long-term allies who remain committed to getting out of Europe believe that Corbyn still shares their faith, but has conceded on Europe in order to keep the party together. Those of his allies who have made the journey to pro-Europeanism believe Corbyn has made it with them. And among his critics, the question of whether his change of heart is genuine sorts the pessimists from the optimists. Those who believe Britain will vote to leave think that Corbyn has played a skilful game of keeping his pro-European supporters onside, but will play a minor role in the referendum’s final stages. Those who think Britain will stay believe the conversion is real – and that when David Cameron breathes a sigh of ­relief on 23 June, he will be forced to raise a glass to his Labour opponent.