Theresa May is going back to Brussels: she might get something, but not enough for her backbenchers

What can the prime minister expect as she tries to reopen the negotiations on the Brexit deal? Remainers in the UK are too quick to take at face value protestations by the EU side that the withdrawal agreement has already been agreed and cannot be changed.

Of course the EU would change it if there were minor adjustments they thought could help to get it through the House of Commons. EU leaders don’t want the UK to leave without a deal, and they want to be helpful to Theresa May if they can. But the problem for the prime minister is that they don’t really believe she is prepared to leave without a deal, and they know that there is a majority in parliament opposed to a no-deal Brexit, as was confirmed last night.

So it will be a tactical negotiation on both sides. Each will be happy to talk up the significance of any changes, but the changes themselves are likely to be cosmetic.

Jeremy Corbyn had some fun at prime minister’s questions today by asking her what the “alternative arrangements” for the border with Ireland, for which MPs voted yesterday, might be. She pretended to read out a list. Options included, she said, a unilateral right to exit the backstop – to which everyone knows the EU cannot agree.

Another option was a time limit on the backstop – to which the same objection applies. So she was left with the third item on her list: schemes for the “mutual recognition of trusted traders”.

No wonder Corbyn replied: “None of that was very clear to me – I don’t know about anyone else.” It is unusual for a politician to admit that they don’t understand something, but on this occasion the Labour leader spoke for the nation.

However, it is in the boring details of trusted trader schemes that some kind of movement in the Brexit negotiations might be possible. It has been reported that Olly Robbins, the civil servant who is the prime minister’s chief Brexit negotiator, has drawn up nine options for changing the agreement on the Irish border, and I suspect that a lot of these will be technical options for handling practical questions of border checks.

Viewed from Theresa May’s perspective, the key is the Democratic Unionist Party. It not only sustains her government; it is in turn the key to winning over many of the hard-Brexit Tory MPs who dislike the backstop for the same reasons. The DUP hates the backstop because it applies some rules of the EU single market to Northern Ireland without applying those same rules to the rest of the UK. As unionists, DUP MPs think that starts to separate Northern Ireland from Great Britain and is a slip on the slope towards a united Ireland.

So what matters to the DUP is less the idea of being trapped in the backstop – the so-called vassalage that gets Jacob Rees-Mogg so excited – than the problem of Northern Ireland being treated differently from the rest of the UK. Never mind that the DUP insists on Northern Ireland’s right to have different laws on abortion and gay rights; it doesn’t want to be treated differently on trade, and it was unimpressed by Michel Barnier’s attempts last year to “de-dramatise” the checks that would be needed to maintain different regulatory regimes in Northern Ireland and Great Britain.

However, Barnier, the EU chief negotiator, did say that those checks could be carried out away from the ports and airports. This is where trusted traders come in: big companies that send goods from Great Britain to Northern Ireland could do all their paperwork (or digital tracking) without the need for physical checks on arrival in Northern Ireland. This is what the DUP calls a border in the Irish Sea, and they don’t like it – except in the case of live animals, which are already subject to checks because of all-Ireland controls on diseases.

So if the prime minister can do a new deal, it could well be something technical of this kind. Corbyn may not understand it, but if the DUP can be persuaded that Northern Ireland will be treated the same as the rest of the UK, Theresa May will be significantly closer to getting her deal through parliament.