Post-Brexit Britain, the Ireland border and frictionless EU trade are dimensions of a question without an answer. The government is now approaching the uncomfortable choice which was inevitable when Theresa May promised to Brexiteers a fully independent trade policy, to the Irish government no hard border on the island, and to the DUP (which provides her Commons majority) that Northern Ireland would leave the EU on exactly the same terms as the rest of the UK.

There are still attempts to deny the choice. Dominic Lawson, writing in the Times over the weekend, suggested that Switzerland, outside the customs union, has the kind of frictionless border with the EU which would solve the Irish problem. He neglected to mention that Switzerland is a member of EFTA, the European trade body, and of Schengen, the European common travel area. The Swiss accept most Single Market rules (with no say over them) and – in practice – free movement of people. The Swiss-EU border has hard infrastructure – cameras, lorry parks, armed guards in places – of the kind the UK has committed to avoid in Ireland.

In reality, this is a binary issue: you are either in or out of a customs union. ‘In’ means the UK applying the same tariffs on imports as the EU and limiting its options for independent trade deals. ‘Out’ means a hard border with Ireland, whether on the island or across the Irish sea, and customs controls on goods moving between the UK and the EU.

However, as evidence has grown – including from the government’s own impact assessments – about the limited extent to which future free trade deals could mitigate damage to trade with the EU, leaving the customs union has become more totemic to Brexiteers.

This has become a definition of Brexit, and is now purely about the politics. There is no majority in the House of Commons to leave the customs union. The government advertised their own lack of confidence in March, by suspending Bills to which Remain-inclined Tory backbenchers had tabled amendments to stay in, rather than risk a defeat.

This week’s vote on a motion proposed by the Liaison Committee – a cross-party grouping of select committee chairs that has been jokingly referred to as the new Centre party – may show how many Tory MPs are prepared to vote against the government on the custom union issue. But, as a non-binding vote, it can do no more than fire a warning shot to the Whips. This is why Nicky Morgan made it clear today that MPs are preparing ‘to show the Government the balance of opinion in the Commons away from the heat of amendments to bills.’

The real nightmare for the government is a binding vote in the Commons against a backdrop of crisis in the Brexit talks. That would embolden soft Brexit Tory rebels; Nicky Morgan and Sarah Woollaston are two of those to watch.

Amendments to the Trade or Taxation (Customs) Bills, likely to be debated after local elections next month, could prove the flashpoint. Tory Brexiteers are pushing the PM to make those votes a matter of confidence. The Fixed Term Parliament Act has made that constitutionally more complicated. But a defeat on the question of the customs union would arguably constitute a fatal loss of Mrs May’s authority.

If the government ends up committed by Parliament to staying in a full customs union, No10 would expect resignations from the Cabinet; perhaps even enough Brexiteers sending letters into the 1922 committee calling to trigger a leadership challenge.

How could the government avoid that? Unless Theresa May is prepared to risk a breakdown of the wider Brexit talks – so far she has shifted UK red lines rather than risk a crisis – there are only two real options.

Firstly, she could at least defuse the risk of a Brexit talks crisis in June over Ireland by agreeing a version of the EU’s ‘backstop’ to avoid a hard border – effectively keeping Northern Ireland in the customs union, and much of the Single Market. However packaged, this would infuriate the DUP; they might withdraw support for the Conservatives in Westminster. It doesn’t, by itself, change the fact that the customs union has become a proxy for a ‘real’ or ‘soft’ Brexit for the respective wings of her party.

Secondly, she could stretch hard Brexiteers’ tolerance by proposing a slower transition out of the customs union. The Treasury, among others, want the UK to stay in a new EU customs partnership, at least until the technology to soften border checks is available. During that time, the UK would be able to negotiate and sign its own trade deals (though not bring them into force). A Parliamentary vote could even be offered on the eventual transition. That might be enough to squeeze the various, essential Brexit Bills through without losing a vote on the customs union. But it may be a bridge too far for some of the hard Brexiteers in Cabinet – and might not buy off potential rebels who see this as their main chance to soften Brexit before the final deal.

Or the government could do nothing, other than twist the arms of potential backbench rebels. Then they risk losing control of the situation.

The choice is imminent.