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As it was always going to, Brexit wrought havoc on Britain’s two main parties in the local elections, with the Conservatives losing more than 1,300 seats on their 2015 total and Labour – who been expecting to make gains – also suffering a substantial net loss.

The big winners were the remain-supporting Liberal Democrats and the Greens as voters expressed their dissatisfaction with Brexit dithering and delay. The Tories lost most heavily in areas that backed remain; Labour in areas that voted leave.

In response to their parties’ trouncings, both Theresa May and Jeremy Corbyn said the result showed voters were fed up with the deadlock and the parties must now reach a compromise. It was a “simple message”, May said: “Just push on and deliver Brexit.”

There was hope for an agreement on some kind of customs union arrangement with the EU after Brexit, the foreign secretary, Jeremy Hunt, told the Guardian; health secretary Matt Hancock also hinted a customs union compromise was possible.

Unsurprisingly, the possibility of a stitch-up by the two party leaderships enraged many of their own backbenchers. The former Conservative leader Iain Duncan Smith said Theresa May must resign or be forced out, while at least 60 Labour MPs have told Corbyn they won’t back a customs pact without a people’s vote.

The shadow chancellor, John McDonnell, subsequently poured cold water on prospects for a deal, saying he did not trust the prime minister. And in Brussels, no one is even remotely confident anything will come of it.

Nothing, once again, has changed.

What next?

That’s the £350m-a-week question nobody can answer. May and Corbyn reportedly hope to agree on a commitment to negotiate a customs union with the EU covering goods that would last until 2022, when the next general election is due.

There would be no independent trade policy in goods until then and, in the run-up to the election, everyone would be free to campaign for their own preferred future relationship with the bloc. This, the Conservatives and Labour apparently hope, might persuade enough MPs from both parties to vote for the deal.

But there are serious doubts that enough Labour MPs in particular would swallow this, and even more serious doubts that the two parties’ competing visions of the post-Brexit relationship are a) acceptable to a majority in parliament, b) technically feasible and c) actually negotiable with the EU.

Britain, in short, is still negotiating with itself. When the EU27 agreed to extend the article 50 deadline until 31 October, Donald Tusk, the president of the EU council, begged the UK not to waste the time. For the moment, that’s precisely what it is doing.

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In the Guardian, Matthew d’Ancona says the whole idea underpinning any possible deal between Theresa May and Jeremy Corbyn is nonsense on stilts:

The Tory right would vote en bloc against what Jacob Rees-Mogg has already condemned as ‘an attempt by the political establishment to avoid Brexit, to have a pretend Brexit’. A great many Labour MPs would oppose a pact with the Tories on principle. On both sides of the house there would be a (perfectly legitimate) fear of the public’s perception that the whole venture was precisely the sort of elite scam that they voted against in 2016. A deal that nobody opted for in the original referendum, brokered by two flailing party leaders in a desperate bid to save their skins: who wants that? Who could possibly be satisfied by that? The public are angry about Brexit already. It’s time to give them a final say.

And Rafael Behr reckons that, with the delay to Brexit and no more negotiations with EU, Westminster has lost touch with reality and retreated into fantasy again:

Who would bet on a tidy resolution before Halloween? There would need to be a dramatic shift towards sober assessment of Britain’s strategic place in the global order, and the available terms of a future relationship with the EU. Debate is moving in the other direction. A dominant portion of the Tory party believes that Britain could comfortably quit the EU with no deal at all. Labour’s position has shrivelled into semantic haggling over a second referendum … The pressure has eased, the deadline has slipped, and the natural tendency to divergence returns. The complex of national demons that British politics conjures when it talks about Brexit drifts ever further away from what the rest of the world means when it talks about the UK leaving the EU.

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On the birth day of a royal baby, a legal expert on Brexit speaks. He is not optimistic: