Stalemate; deadlock; impasse – or as Jeremy Corbyn put it at PMQs on Wednesday, complete disarray. The narrative about Theresa May’s approach to Brexit and the customs union has barely changed for weeks.

Yet officials were congratulating themselves quietly on Thursday about making incremental progress on a closely interlinked issue – the Irish border.

After media reports overnight, Theresa May issued a carefully worded statement as she arrived in Sofia, Bulgaria, repeating the familiar statement that Britain would be “leaving the customs union”.

But Whitehall sources confirmed that earlier this week, herBrexit war cabinet – more formally known as the strategy and negotiations subcommittee – agreed a fresh proposal that Britain hopes will avoid a hard border in Northern Ireland, if all else fails.

The “backstop” was written into the December agreement between London and Brussels, at the request of Ireland. It commits Britain to “maintain full alignment with those rules of the internal market and the customs union which, now or in the future, support North-South cooperation, the all-island economy and the protection of the 1998 Agreement.”

What that will mean in practice needs to be spelled out in detail. The EU has published its own definition, which would involve Northern Ireland remaining in the single market and customs union – but May told parliament “no prime minister could ever agree” to that.

May and her chief negotiator, Olly Robbins, have won the backing of key ministers for a British counter-proposal.

If the backstop had to be enacted, Britain would agree to maintain the common external tariff – the import tax levied on goods coming into the EU, which is hated by the Brexiters – and align its regulations with Europe’s, for a strictly limited time-period.

But Britain would seek to opt out of the common commercial policy, which prevents member countries from negotiating independent trade deals.

The trade minister, Liam Fox, has been touring the world to lay the groundwork for future deals – though presumably these would only kick in once Britain has stopped applying the external tariff, as Fox will have little to offer before then.

The government hopes this approach will meet the definition of the backstop set out in the December agreement – and that it could help win over Dublin, which would put pressure on Brussels to accept it.

There were signs already on Thursday that the cabinet compromise may shatter on contact with political reality. After May met the Taoiseach, Leo Varadkar, in Sofia, he stressed that he would like to see Britain’s plans in writing – and underlined the importance of single-market membership for avoiding a hard border.

The prime minister is expected to put her plans on paper in the next fortnight, with the hope that it will be enough to secure the backing of the EU27 at next month’s European council to move beyond the Irish border issue.

There are other small signs of progress this week too. David Davis’s DexEU has announced that it will shortly publish a 100-page white paper, setting out what the government hopes to get out of the negotiations.

May’s spokesperson promised on Friday that the EU withdrawal bill, which has been peppered with amendments by rebellious peers, will come back to the Commons within “weeks, not months”.

There had been rumours that May was so afraid of her own backbenchers that she would not hold any contentious votes until the autumn.

None of this changes the fact that there are two conflicting visions of Britain’s post-Brexit customs arrangements within May’s inner cabinet, neither of which is expected to be ready by the end of the transition period in December 2020.

In that case, some in Whitehall believe the backstop could end up being a stopgap, while the details are worked out – or in the case of the Brexiters’ preferred max-fac option, technology is developed.

It is unclear whether Brussels – or indeed the Conservatives’ allies in the DUP – will be prepared to accept this half-in, half-out approach to the backstop. The European council president, Donald Tusk, has signalled that the devil is in the detail.

But at least, for the first time in weeks, May was able to rebut the charge, heard frequently from Brussels, that the Brexit process has stalled because the government is so caught up in internal warfare that it has nothing new to say.