“We’ll get him for this,” a Jeremy Corbyn aide snapped angrily when Keir Starmer defied the Labour leader by telling the party’s September conference that Remain would be on the ballot paper in any Final Say referendum.

The shadow Brexit secretary, deliberately going beyond the speech approved by Corbyn’s office, won an ovation in the hall, which showed where Labour members stand. His defiance, and the furious reaction in Team Corbyn, shows that divisions over Brexit are far from confined to the Conservatives.

Corbyn has done a pretty good job of masking Labour’s splits. His “constructive ambiguity”, designed not to alienate Remain or Leave voters, also helps to hold his party together.

It’s easier in opposition than government, of course. Corbyn can dodge questions that ministers cannot. On Thursday, he rightly criticised the political declaration on future UK-EU relations as “vague” and “waffle”, but Labour’s position is open to the same charge.

Corbyn can hardly be blamed for wanting to let the Tories’ stew in their own juice; why throw your drowning enemy a lifeline? However, he will soon have to come off the fence. His decisions will matter and, rarely for an opposition, could determine the country’s future.

The vast majority of Labour MPs will obey the party whip and vote against Theresa May’s deal in just over two weeks. When it is rejected (it’s a “when”, not an “if”), they will unite in opposing a no-deal exit next March and voting for a general election. But the Tories and Democratic Unionist Party will reject Corbyn’s call for an election. So the really big question is not what Labour does after the first Commons vote, but in the second one held after, as even her own ministers now assume, May returns from Brussels with some cosmetic changes to her deal and stakes all on getting it through as the clock ticks down to March.

A sizeable number of Labour backbenchers, seeing turmoil on the financial markets, feeling pressure from business and fearing a no-deal departure, might then switch sides and rescue May. They would argue that they had put the party first in the first Commons vote, but now had to act in the country’s interests. There are even rumours at Westminster that, if May’s tweaks shifted the deal in Labour’s direction, the Opposition might abstain, allowing the agreement to win a Commons majority.

I doubt that would happen – Corbyn would face a revolt from party members, and many Labour MPs would still vote against the deal. But I suspect Corbyn might not lose much sleep if 30 to 40 Labour MPs backed a revised deal and the UK left the EU on schedule on 29 March. “Constructive ambiguity” survived an election last year but would not survive another referendum. Corbyn would be relieved to see the back of Brexit, and return to the comfort zone of domestic policy.

Labour’s conference agreed that, if May’s deal and an election are rejected, the party would “support all options remaining on the table, including campaigning for a public vote” (Team Corbyn didn’t want to call it a “people’s vote” because the People’s Vote campaign has several refugees from Corbyn Labour on its staff, and is seen by Labour figures who oppose a referendum as an embryo centre party).

It is obvious from his words and body language that Corbyn does not want a referendum. “It would kill us in the north,” one ally said, predicting a backlash among Labour supporters who voted Leave in 2016. But there is growing evidence that opinion has changed since then, as the cold reality of Brexit bites.

Jeremy Corbyn say he does not know how he would vote in fresh EU referendum

On the eve of Labour’s conference, Corbyn promised he would be “bound” by its decision. After championing internal democracy, he could hardly say anything else. But some Labour members claim his recent media interviews – he told Der Spiegel Labour could not stop Brexit – were “not aligned with party policy” and are urging him to keep his promise.

One route for Labour would be to join forces with other opposition parties and some Tories in supporting a “Norway option” of European Economic Area membership, topped up with Labour’s policy of a permanent customs union. MPs in all parties believe there is a natural Commons majority for this. It could pass Labour’s deliberately flexible “six tests” if the leadership wanted it to.