The British media, unusually consistent in their analysis of what happened, agreed that Theresa May was humiliated in Salzburg last week. She had come armed with her “Chequers” plan for the UK’s long-term relationship with the European Union. Chequers is the official country residence of British prime ministers; located 50 kilometers from London, it’s where the plan was thrashed out by May’s Cabinet in July.

May never expected other EU leaders to welcome the Chequers proposal with open arms. She did, however, expect them to accept it as a starting point for the final round of negotiations. Instead, last Thursday, Donald Tusk, president of the European Council, said bluntly that the plan “will not work.” He demanded a new approach from the UK; otherwise, the EU would terminate the Brexit negotiations at next month’s Council meeting. The widely expected extra meeting in November to finalize a UK-EU deal would not happen.

The following day, May, bruised and angry, blamed other EU leaders for not showing “respect” for the UK position. She demanded a different tone for the talks over the next few weeks. She complained that EU negotiators had attacked her plan, but without proposing any alternative.

May’s real problem, however, has nothing to do with tone or respect. The gulf between the UK and EU positions cannot be bridged by a soothing form of words. Whatever language is used, and however friendly or unfriendly the mood of the talks over the next few weeks, one side or the other has to blink: that is, to surrender one of the redlines it says cannot be violated.

The underlying issues, which mainly concern trade, are especially fraught because of the collision of politics and economics at the only land border between the UK and the rest of the EU, between Northern Ireland and the Irish Republic.

These are May’s redlines:

• The UK must leave the Customs Union, so that it can negotiate its own trade deals with the rest of the world.
• The border between Northern Ireland and the Irish Republic must remain completely open, as it is today, with no physical infrastructure, checks, or border posts of any kind.
• In terms of customs arrangements and business regulations, Northern Ireland must remain fully part of the UK, with no barriers, tariffs, or checks impeding trade between Northern Ireland and the British mainland.

This is the EU’s position on these issues.

• If the whole of the UK leaves the Customs Union, then goods being exported from the UK to the EU must be checked to confirm that they conform to EU regulations. (This, incidentally, would apply to a Canada-style trade deal currently being proposed by some Brexiters in London, which is essentially about reducing tariffs to zero, not abolishing non-tariff barriers such as product standards. Border checks would remain.)
• Border checks would have to be made at all places where UK-EU trade takes place, including the border between Northern Ireland and the Irish Republic. For the UK to leave the Customs Union and keep the Irish border completely open, Northern Ireland must remain in the Customs Union, while the rest of the UK goes its own way.

Shortly before last week’s Salzburg meeting, Michel Barnier, the EU’s chief negotiator, proposed one way forward: keep Northern Ireland in the Customs Union, and have a smart system for checking trade across the Irish Sea to minimize the checks on goods being traded between Northern Ireland and the UK. The UK rejected this on principle; for however unobtrusive the checks, this plan would violate the principle that Northern Ireland must not in any way operate according to different rules than mainland Britain.

So where do we go from here? Logically there are five possibilities. The first three are ways in which Theresa May might give way.

• The UK agrees to remain in the Customs Union and gives up the right to negotiate its own trade deals with the rest of the world.
• Reimpose checks and border posts along the 500 kilometer border between Northern Ireland and the Irish Republic.
• Allow Northern Ireland to remain in the Customs Union while the rest of the UK leaves it, and establish some kind of system for checking goods crossing the Irish Sea.

It is hard to see May agreeing to any of these options and keeping her party together. Indeed, doing any of these things could lead to her being deposed as the Tory Party leader and British prime minister.

The other two options would involve a retreat by Barnier and governments across the EU:

• Allow the UK to leave the Customs Union and no longer be bound by its rules and enforcement mechanism (ultimately via the European Court of Justice), but nevertheless trust the UK to enforce EU standards in goods, and allow frictionless trade to continue between the UK and EU; or
• Impose checks on UK-EU trade, but do so across the Irish border using smart technology in a way that requires no border posts or physical infrastructure.

The UK’s preferred solution is the fifth and final option. The trouble is that the technology does not exist at the moment. Maybe it will be developed in some years’ time—five, ten, or perhaps twenty. But until then, if the UK is not to crash out of the EU without a deal, one of the other four options must be agreed.

At present, May is fiercely opposed to all three of her options—and might not survive politically were she to change her mind. Equally, the chances are vanishingly small of the EU allowing the UK to enjoy the benefits of Customs Union membership without agreeing to abide by its rules and system for enforcing them.

The current deadlock, then, is rooted in a set of real, apparently unbridgeable, differences between the two sides. Sharp language and complaints of disrespect don’t help, but they are not the real cause of the current standoff. Unless someone sacrifices one of their redline principles, the chances of the current Brexit negotiations ending without a deal look high and rising.