The proposal might sound bizarre but it is the only way May can get her deal through parliament and survive as prime minister. Her challenge is, and was always going to be, a matter of arithmetic. Yesterday saw disgust from all sides of the House directed at May’s negotiated deal, but the fact is that no kind of Brexit deal can command majority support from MPs. Her only sensible option now is to offer Labour a general election immediately after Brexit, in return for backing her deal.

If May produced a Canada-style option, sought by hard Brexiters, it would be defeated by Labour plus her own pro-European Union rebels. A Chequers-style deal – the present plan – would be defeated by Labour, the DUP and Tory hard Brexiters. Many people believe that May should therefore go for something even softer, such as the European Economic Area (EEA) option of remaining in the single market and customs union, to win over Labour MPs. But this option would also be defeated. An EEA-style deal would face immense backlash from Tory MPs, most of whom have spent the last two years (on May’s instructions) arguing that remaining in the single market fails to respect the referendum result.

The EEA option may be what many Labour MPs desire but Corbyn would undoubtedly whip his party to vote against anything May produces – even if it were directly lifted from Labour conference resolutions. This is because the Labour leadership’s overarching aim is not to stay in the single market but to force a general election and take power.

Furthermore, and inconveniently for May, the Labour MPs who care least about Corbyn’s whip happen to be those who are most ardently pro-Europe. The likes of Chuka Umunna, Yvette Cooper and Chris Leslie will vote down anything that doesn’t sound like another referendum.

This is why May should offer the Labour leadership an election in April, immediately after the Brexit deadline in March, in return for Labour backing her deal through parliament. Forcing a general election has been Labour’s raison d’être ever since their impressive performance in 2017, when Corbyn first realised he wanted to – and could – be prime minister.

But perhaps most importantly, forcing an election is at the heart of Labour’s surprising level of unity because it allows them to fudge their stance on Brexit: it avoids committing to any particular model of Brexit, allows the second referendum option to remain on the table, and ultimately means Labour has retained Remainers without losing too many Leavers.

Negotiating a deal and getting it through has been May’s personal raison d’être since she became leader. Perhaps surprisingly, she seems to genuinely believe her deal is best for the country (“with every fibre of my being”), but on a more pragmatic level her fate depends on it. Even without an obvious successor, Conservative MPs would not hesitate to depose her if she failed to get her deal through parliament. Like it or not, she needs Labour’s support, and she might find surprising common interest with the Labour leadership.

From Labour’s perspective, it is better to have a guaranteed election in April than risk a no-deal scenario to get one now. Furthermore, despite the facade of unity displayed at conference, the fact is that having, and even winning, a general election before March would be a nightmare for Labour.

As with the Conservatives, Labour is too divided on Brexit to run on a coherent platform, let alone conduct negotiations. Prime Minister Corbyn would presumably back Brexit with a majority of MPs who are, let’s face it, still Remainers, now emboldened by a shift in political mood. And that’s not to mention the fact that Brussels would, in all likelihood, make a near-identical offer to Labour’s negotiating team as they did to May’s, thus leading to the same sort of parliamentary gridlock.

Corbyn’s ideal world is surely to be prime minister without having to make the big decisions about Brexit. An April election would give a new Labour government an opportunity to influence the future UK-EU relationship without pressure to stop Brexit altogether.

A few questions remain. Could May guarantee an election in April, if it were promised to Labour? She would need two-thirds of MPs’ support, which should be doable – just half of her party would need to back the idea plus opposition MPs who would leap at the opportunity.

Is Labour guaranteed to win it? This is much less clear. They certainly seem optimistic about winning an election, but if May successfully pushed through a deal and Brexit went ahead relatively smoothly, she might find favour with voters. Regardless, an April election is far more in May’s personal interests than an election now, when she is unpopular with both her party and voters.

The groups that would stand to lose from this arrangement would be arch-Remainers who want a second referendum, and hard Brexiters who want to remove May and push through “Canada-plus-plus”. But these groups are too ideologically opposed to cut their own deal, and neither group has control over their respective parties. The opportunity falls to the leaders of the two main parties to cut their own deal, which, perhaps surprisingly, would benefit them both at the expense of their respective fringes.