The informal summit in Salzburg had been meant to be – at the very least – a stay of execution for the prime minister. A time when EU leaders offered Theresa May a few insincere air kisses and told the world's media that her Chequers Brexit proposals contained one or two promising ideas. Instead May was given a lesson in brutality with EU Council president Donald Tusk declaring her plans to be completely unworkable.

An hour after being given the bad news in person, the sweaty, panic-struck prime minister stepped into an airless sideroom to give her own version of events. Nothing had changed, she insisted. Something we had heard before during last year's general election campaign when she tried to persuade the country that dropping an intended dementia tax, just days after including it in the Tory party manifesto, was not a policy reversal.

No matter that the EU had just ruled out any Chequers deal. No matter that large numbers of her own MPs had already ruled out any Chequers deal, making it a mathematical impossibility to get it through parliament. According to the prime minister, her Chequers deal was still the only credible deal on offer and she was going to carry on regardless. It was an extraordinary piece of political theatre: an exercise in denial that was a personal humiliation right up there with her party conference speech last year.

Quite how May managed to so badly misread the mood of the EU summit is still not entirely clear. Leaked accounts suggest that the EU leaders hadn't appreciated the tone of the British prime minister's remarks to them both during her brief post-dinner speech or in private one-to-one conversations. Not for the first time, May's notorious tin ear came back to bite her big time as the EU decided it was time for some straight talking of its own. The UK government's repeated claims that it had fallen over backwards to offer concessions while the EU had basically done nothing but say no to everything needed to be ended once and for all.

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It had been the UK that had chosen to leave the EU – not the other way round – and, as such, it was up to the UK to make up its mind what kind of future arrangement it wanted. The EU had always been very clear that there were only two basic options available – the hard Brexit of a Canada-type deal or the soft Brexit of the Norway model, in which the UK would effectively remain inside the customs union and the single market. It really wasn't the EU's problem if the UK was not that thrilled with either choice: there would be no cherry picking on offer. The UK would have to make up its mind what was more important: regulatory alignment or a hard Irish border.

In yet another bizarre twist, May chose to double down on her Salzburg press conference by giving a televised statement from Number Ten the following day in which she was again adamant that her Chequers deal was the only credible Brexit solution on offer. It was like watching someone sticking their fingers in their ears and shouting “La, la, la” to drown out any dissent. She also – in a desperate attempt to channel her inner Churchill – said it was time the EU started showing her more respect. She obviously hasn't been paying attention to the open contempt with which many of her MPs have been treating her for the last year.

So, with six months until the UK is due to leave the EU, the country is now effectively no nearer agreeing a deal than when we triggered Article 50 in March 2017. The prime minister's proposals are dead in the water and the Eurosceptics in the Tory party are no nearer to coming up with anything remotely workable as all their suggestions involve fantasy economics and imaginary solutions to the Irish border problem. The prospect of a no deal would seem to be increasing by the day – except for the fact that it seems impossible to believe that parliament would vote for that either.

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With all this chaos going on, it would be some comfort if Labour had some solutions of its own. But judging by some of the fudging going on at its party conference in Liverpool this week, Labour too is stuck between a rock and a hard place. And guilty of its own double standards. The overwhelming majority of Labour's members want a "People's Vote" on the final deal. Normally the leadership has been only too happy to give the Labour grassroots exactly what it wants – the moves to make it easier to deselect MPs being a case in point – as their wants closely coincide Jeremy Corbyn's.

But not in this case. Not only does the Labour leader have well known Eurosceptic tendencies, his leadership team believe it will be electoral suicide to wholeheartedly back a People's Vote. Labour can't afford to alienate the working class Leave voters in the Midlands and the North if it's to win any general election and fear that a People's Vote would be seen as a betrayal of the referendum result. With this in mind, Labour has come up with a Brexit plan that is almost as meaningless as the Tories': to call for a general election and, if that is not forthcoming, to pursue “all options for a meaningful People's Vote”. A promise vague enough to mean everything and nothing. It could mean a chance to either vote for whatever deal the government cobbles together or a no deal or it could be a chance to rerun the referendum. No one knows. Least of all Labour. At a time when clarity is demanded, none is forthcoming.

So it feels as if the whole of the UK is effectively being used as pawn in a chess game, being played between Conservatives and Labour, neither of whom have any real idea what to do and are reduced to circling one another, hoping the other blinks first and makes the fatal mistake. It's hard to see how this ends well for anyone.

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