There is no political crisis so bad that Donald Trump can’t somehow manage to make it worse.

And so it has come to pass with Brexit. Just as Theresa May is straining to persuade the nation that her compromise deal isn’t as bad as it sounds, our erstwhile ally pops up to pour a bucket of cold water all over it. Britain “may not be able to trade with” America in future, the president declared, although he presumably means it may not be free to strike a trade deal, rather than that all goods will stop at the border.

It’s going to be harder to talk him off the ledge this time, without the incentive of a photo opportunity with the Queen

Downing Street insisted he was wrong, and that Britain would be free to strike trade deals once the transition period was over, but still, the damage is done. It seems President Obama was right to warn that Brexit would put us at the “back of the queue” for a trade deal with the US, although naturally that’s not quite how Brexiteers see it; for them, it’s proof not that Obama should have been heeded but that May must have got the wrong deal. It couldn’t possibly be their fault, because nothing ever is.

Usually, this kind of embarrassingly public divide between two allies would indicate that someone in Downing Street had taken their eye off the ball and neglected to square it with the White House. After all, it’s not as if Trump didn’t give fair warning; when he visited Britain in July, he told the Sun that a soft Brexit like the one that has just been negotiated would probably “kill” a trade deal.

But since this is Trump, and even his own staff have a hard time keeping him on message, perhaps it was never realistic to expect him to slot neatly into No 10’s communications strategy.

The first priority for Downing Street will therefore be trying to limit the fallout. Back in July, Trump was persuaded into half-recanting by calling the Sun story “fake news”, even though he was on tape saying exactly what he had been reported as saying. It’s going to be harder to talk him off the ledge this time, without the incentive of wanting his photo opportunity with the Queen to be a success, but presumably someone will have to try.

The second question, given that Trump clearly isn’t taking his lines from the woman whose hand he once so excruciatingly held before the cameras, is who he might be listening to instead. Last time he embarrassed May over Brexit, suspicion fell on Boris Johnson, who had made it his business as foreign secretary to get close to the new White House regime. Trump’s suggestion that this is “a great deal for the EU”, but not for Britain, certainly dovetails closely with the Tory rebels’ line. But the president also famously sets great store by the judgment of Rupert Murdoch, among others, and has long had a soft spot for Brexit based mainly on the fact that the referendum result reminds him of his own victory against the establishment.

The third and most intriguing question for Downing Street, however, is how seriously this threat should be taken. Just as Obama’s warning backfired in some ways against the remain campaign, irritating leave voters who resented his interference, there will be plenty of voters who can’t stand Trump and never wanted to eat chlorinated American chicken anyway. If anything, some will find the fact that he hates this deal reassuring, interpreting it as a fit of pique at a missed opportunity to get Britain over a barrel.

But his words are still a brutal reminder that, in the unlikely event May’s deal limps through parliament, the hard work of building viable trading relationships outside the EU will still only just be beginning. It was always obvious where the bargaining power would lie in trade talks between a small country with big ideas and any global superpower, and it was wilfully naive not to see that even friendly nations would try to exploit our self-inflicted crisis to extract a better deal for themselves. That doesn’t stop at trade, either; British freedom to criticise human rights abuses or controversial decisions by governments with whom we’re seeking deals may also be curtailed.

All Trump has really done is remind everyone of that painful reality. So much for the idea of taking back control, or reclaiming sovereignty from Brussels, if all we end up doing is surrendering it to Washington or Beijing.

• Gaby Hinsliff is a Guardian columnist