Why doesn't Theresa May's counter-offer on the rights of European citizens living and working in Britain pass muster among the EU27? It all comes down to one of the biggest sticking points in the Brexit talks: the role of the European Court of Justice.

The European Commission, under direction from the leaders of member states, wants the rights of the three million living here and of the British diaspora in the EU guaranteed by the European Court. Why? Because that way, the status of EU citizens here or that of British nationals in the EU aren't subject to the whims of a simple majority vote in the legislature.

This is where Liam Fox, as crassly he might have put it, has a point about the difference between the UK and the EU27, being that the UK does not "need to bury" its 20th century history. We're one of the few countries in the EU where political elites get away with saying, "Well, what's the worst that could happen?" when it comes to checks on legislative power. For the leaders of member states, a guarantee not backed up by the European Court of Justice is no guarantee at all.

That comes down to the biggest sticking point of the Brexit talks: rules. In terms of the deal that most British voters, Leave or Remain, want – a non-disruptive exit that allows the British government to set immigration policy – UK politicians can get that, provided they concede on money and rules, ie we continue to follow the directions of the European Court while having no power to set them. Britain could even seek its own trade deals and have that arrangement.

But the problem is that deal runs up against the motivations of the Brexit elite, who are in the main unfussed about migration but are concerned about sovereignty – and remaining subject to the rule of the ECJ without being able to set its parameters is, it goes without saying, a significant loss of sovereignty.

Can a fudge be found? That the Article 50 process goes so heavily in favour of the EU27 and against the leaving member means that the appetite on the EuCo side for a fudge is limited.

But there is hope, as David Davis has conceded that there will have to be an international guarantor, as of course there will have to be. If you trade across borders, you need a cross-border referee. If a plane goes up in one country and lands in another, then it is, by necessity, regulated across borders. (That arrangement has also been mooted by Sigmar Gabriel, foreign minister in Angela Merkel's government. But that Gabriel's centre-left party looks likely to be expelled from coalition after the next election means that his support isn't as valuable as many Brexiteers seem to think.)

On the Conservative side, a new EU-UK international body would satisfy the words of May's ECJ red line. On the EU27 side, that the body would, inevitably, take its lead from the treaties of the EU sans Britain and the ECJ would mean that in spirit, Britain would be subject to the ECJ by another name.

But it comes back to the Brexit dilemma. You can satisfy the voters' demand for non-disruptive control of British borders. You can satisfy political demand for sovereignty. But you can't have both. May – and whoever replaces her – will face the same question: who do you disappoint?