Under pressure from France's president Macron, the Brexit delay to 31 October is shorter than Donald Tusk, the EU's president, and many government heads thought desirable - though still considerably longer than Theresa May consistently said was acceptable.

Its impact may well be to turn the Tories into the no-deal Brexit party and Labour into the referendum party, via a change of Tory leader and even a general election.

Here's how and why.

What was agreed late last night poses an immediate and important question for MPs and ministers, because there is an explicit opportunity for the UK to avoid participating in the EU election by leaving without a deal on 1 June.

This is how the EU council conclusions put it:

"If the UK is still a Member of the EU on 23-26 May 2019 and if it has not ratified the Withdrawal Agreement by 22 May 2019, it must hold the elections to the European Parliament in accordance with Union law. If the United Kingdom fails to live up to this obligation, the withdrawal will take place on 1 June 2019".

For those Tory Brexiters who would prefer that the UK leaves without a deal, this represents a tantalising prospect; they might even see it as the most generous offer the EU has made to date.

It also puts the PM in a difficult spot - given that she has said she could not "as prime minister" consider a Brexit delay beyond 30 June.

May will use the deadline to put pressure on MPs to agree the Withdrawal Agreement and the associated legislation by 22 May, so that in practice the UK would Brexit on 1 June with a deal, in a managed and controlled way.

But she knows - as Labour's shadow chancellor John McDonnell confirmed on my show last night - that talks with Labour on a compromise Brexit deal are not going well enough for her to have any confidence her deal can be approved in time.

McDonnell himself told me he thought it impossible all the necessary legislation could be passed by the end of June, let alone the end of May - and even in the highly unlikely event that an entente could be reached between opposition and government (the talks so far have not resulted in agreement on anything of substance, McDonnell said).

And just in case you thought McDonnell was posturing, the foreign secretary Jeremy Hunt gave me a forensic critique of the central element of the Brexit that Labour wants, namely membership of a customs union, and he could not reassure McDonnell that Theresa May's successor - who could be Hunt himself - would honour any pact that May reaches with Corbyn in the coming days.

So the probability of a cross-party Brexit pact is so close to zero as to be barely worth considering.

Which will focus minds urgently on all sides of the Brexit argument.

Tory Brexiter ultras, in the European Research Group and in the Cabinet, will urge the PM to stay true to her refusal to countenance a delay beyond 30 June and prepare for a no-deal Brexit on 1 June.

Presumably she will refuse, given that she - and all of us - know that a majority of MPs oppose a no-deal Brexit on that or any date, and will find a way to block it.

So the more zealous Conservative Brexiters will flirt with the idea of evicting Theresa May immediately and replacing her with a prime minister who might somehow find a way to cancel the EU parliamentary elections, create a bona fide EU crisis and thus extricate the UK from the EU without a deal on 1 June.

Support for such a coup would run wider among Conservative MPs than you might expect, largely because the European elections are likely to be a catastrophe for Tory candidates.

But in practice such a coup looks more Quixote than Guevara. It would be more rational for the PM's Tory critics to see All Hallow's Eve, 31 October, as their target date for a no-deal Brexit, so that a leadership election and even a general election could be held not quite so hastily and could provide a proper mandate for such an abrupt break with the EU.

And were that to happen, and I think it might, the corollary - again implied by McDonnell last night - is that Labour could soon become the official party of a people's vote, a referendum.

As a result of last night's Brexit delay, the time may not be far off when Brexit politics becomes slightly simpler (thank goodness) - with Tories becoming the "hard" Brexit party, and Labour the referendum party.

And for the minority of Tory and Labour MPs who will hate this stark repositioning, there is a life to be found in new parties or outside politics.