As the Salzburg summit broke up amid a slew of accusations and counter accusations, it was easy to imagine that, with only six months to go until the UK is due to leave the EU, we are as far as ever from securing the deal that both sides want. It may be, however, that the Brexit negotiations themselves are the least of the problems confronting those attempting to ensure the UK leaves the EU in an orderly manner.

Whatever deal is agreed must be approved by the British parliament. And it’s interesting to note that the EU has been discussing plans to step up its no-deal arrangements, partly because of a fear that the House of Commons will prove unable to approve any conceivable Brexit outcome.

The first thing that should be said is that, as our new UK in a Changing Europe report underlines, the process of parliamentary approval of the Brexit deal will be anything but straightforward. First up, MPs will be presented with a complicated pair of texts (the withdrawal agreement and a political declaration on the future relationship) and asked to vote on them quickly.

The Fixed-term Parliaments Act deprives the government of the power to do what John Major did in 1993 and make the vote one of confidence, hence removing an important parliamentary weapon from Theresa May’s arsenal. MPs could reject the deal and then support her in a vote of no confidence.

And, something that is often forgotten, parliament will have two bites of the Brexit cherry. Not only will it have to approve the withdrawal agreement itself, but it must also then vote on the bill that makes it law in the UK. There is good reason to suppose that, with time to mull over the details of the Brexit agreement, and with more detail on the practical arrangements the government intends to put in place for, say, Northern Ireland, MPs might vote one way in the first vote and another in the second. After all, when the UK joined the EU, the initial agreement passed comfortably while the 1972 European Communities Act that brought membership within the ambit of UK law barely squeaked through.

‘It will be interesting to see to what extent Michel Barnier colludes in presenting the choice as one between the deal and no deal at all.’ Photograph: Aris Oikonomou/AFP/Getty Images

And while the process is perplexing, the politics is still more so. Frankly, anyone who claims to know how parliament will vote on the Brexit deal(s) is kidding themselves. The fact is that many MPs are not certain which lobby they will walk through. So we are left to read the runes, and interpret the gnomic utterances of our political class.

Take the Labour party. On one level, the leadership achieved what it wanted from party conference this week – something for everyone on Brexit. While Keir Starmer surprised himself by inspiring a standing ovation for hinting at possible support for remain in a new referendum, union bigwigs dismissed the idea of it being an option at all.

Jeremy Corbyn’s offer to support a government willing to deliver the kind of Brexit he now seems to want is less of a game-changer than it may seem, if only because the prime minister simply cannot opt for membership of a customs union. Supporting people on the condition they do something they can’t may not be practical, but might be good politics.

Labour wants to look reasonable on Brexit and move on. Its new party political broadcast indicates its focus is on towns (which voted predominantly leave in the 2016 referendum) rather than on trying to make further inroads among remain voters.

But of course, this is the leadership. The reason why the parliamentary votes to come are so hard to predict is that the leadership on both sides no longer holds sway. Several factors will determine how MPs vote when it comes to the final Brexit deal. First, obviously, content. The “softer” the prime minister goes, the more she will have to hope to attract Labour backbenchers to make up for rebellions among her hard Brexit-inclined colleagues, and vice versa.

Second, timing. It is in the government’s interest to have the vote as late as possible, if only to reinforce its narrative that parliamentarians face a choice between May’s deal and no deal at all. Such a choice, successfully presented, might swing sufficient numbers of MPs who dislike either Brexit or the terms of Brexit offered, to reluctantly give it their support.

Third, the alternative is crucial. While the Fixed-term Parliaments Act prevents May from turning a vote on a substantive issue into a vote of confidence, she and her whips will be anxious to persuade Tory MPs that defeat will jeopardise the survival chances of the government. This, in turn, will enable them to deploy three different threats.

To begin with, Brexiters could be confronted with the possibility that a government defeat would increase the likelihood of a second referendum – and therefore of Brexit not happening at all. “My Brexit or no Brexit” is a mantra they should get used to.

Second, Conservative soft Brexiters would face a choice between whatever May signs up to and no deal. The government will doubtless try to argue that whatever it negotiates is the only deal on the table and there will be no time to come up with an alternative. It will be interesting to see to what extent Michel Barnier colludes in presenting the choice as one between the deal and no deal at all.

Finally, all those Tories who sat at home watching John McDonnell on the telly from Liverpool know how high the stakes are should a failed Brexit unseat the Tory government. Not for nothing did one of their number remark to me recently that Jeremy Corbyn is “the best chief whip we’ve ever had”. No 10 must be counting on the fact that a terror of the Labour leadership will instil some much-needed discipline in a parliamentary party from which, of late, it has been lacking.

So there’s all to play for and not much at all is clear. Once attention shifts from Brussels to Westminster, the real Brexit endgame will begin. And we have no earthly clue how it will play out.

• Anand Menon is director of The UK in a Changing Europe and professor of European politics and foreign affairs, King’s College London