Before the big vote on Tuesday night, the EU's 27 government heads will provide greater reassurances - probably in the form of a collective letter to Theresa May, and within the mandate confirmed at the last EU Council - that the controversial Northern Ireland backstop will not and cannot be forever.

What does that mean?

Well for those MPs agonising about whether or not to support the PM's Brexit plan, and who think the word of political leaders counts for something, a few votes may move in Theresa May's direction.

And maybe, in the words of one senior British minister, May will be able to frame the letter as being both "substantive" and "legally" significant.

But it will not sway the vast majority of her critics, because the so-called Withdrawal Agreement will not be re-opened - and whatever the letter's legal force it could not trump the international treaty that is the Withdrawal Agreement, so there will be no legally binding guarantee that the backstop will fall away by any specified date.

That means the PM still loses on Tuesday night, but possibly by fewer votes than would otherwise have been the case.

These last minute manoeuvrings highlight May's single most important strategic mistake: her failure to construct a majority in parliament for a vision of the UK's future relationship with the EU.

If we knew what kind of commercial and security relationship we would ultimately have with the EU, the backstop would not be the problem it is: there would be widespread confidence on both sides of the channel that if the backstop were used at all, it would be of desperately short duration, for the simple reason that there would be a high degree of certainty about post-Brexit negotiations to put in place alternative trading arrangements that would make the backstop wholly redundant.

But as a senior official from an EU government says, "given principles and red lines on both sides, it is difficult to see what future relationship we could ever agree on".

That is why it is NOT barking mad to suggest that the backstop could stay in force till after we're all pushing up the daisies.

If only May had spent more - or any - time negotiating with MPs from all parties to establish a consensus on an acceptable post-Brexit relationship with the EU, she would stand a decent chance of winning the notorious meaningful vote.

But Jeremy Corbyn's vision of a post-Brexit relationship with the EU is a million miles from that of many of his own MPs and from that of Remainer Tory MPs and even further away from that of Brexiter Tory MPs.

The only thing that unites most of them is their contempt for the backstop. Which is why I cannot find any minister who thinks there is any way she can win on Tuesday - and most expect her to lose big.

What follows the humiliation of that defeat?

Well the foreign secretary Jeremy Hunt is probably correct that Parliament would find a way to block a no-deal Brexit, such that the choice for MPs will boil down to what he calls "a version" of her Brexit deal, or no Brexit at all (via a referendum).

Will the PM have to start negotiating with Jeremy Corbyn? Credit: PA

Should and would this rising probability that we will remain in the EU after all lead the PM to start treating with Jeremy Corbyn, to agree a vision of the UK's future trading relationship that might command a majority in the Commons?

One influential minister told me: "My judgement is that Corbyn would never compromise on any deal because he needs a crisis and general election, so moving to a softer Brexit just loses more Conservative [MP] votes without gaining very many Labour ones."

That sounds like a no to May attempting any kind of serious bridge building with Labour.

What then?

Well ministers assume the PM would have one last go with EU leaders to persuade them - despite all their protestations this cannot happen - to open the Withdrawal Agreement and formalise the temporary status of the backstop, by creating a mechanism that would give MPs more confidence they could terminate it.

But here we return to the Catch 22 of the Prime Minister's own making.

EU leaders are not convinced that if they made what they would see as the mother of all climbdowns, by effectively burying the backstop, Theresa May would even then win her vote - because of that absence of any consensus in parliament about what the future relationship with the EU should be.

The correct view in EU capitals, given by an official from one of them, is that Brexiter and Remainer MPs reject the Withdrawal Agreement for several reasons: "The backstop is not the only issue for both these groups."

Unless May can persuade EU leaders that she would win the vote on the back of a climbdown on the backstop, they will not put themselves through the humiliation of such a retreat.

We are heading, as the PM warned, for uncharted waters. And Theresa May - some would say - only has herself to blame.