Peter Mair once observed a curious paradox in European elections: people often use their votes to express their dissatisfaction with the fundamental nature of the European Union, despite that being outside an MEP’s purview – the Union is founded on treaties signed by national governments. Conversely, national governments are often elected to pursue policies that are properly the domain of the European Parliament, and so find themselves unable to deliver on their promises – an effect especially pronounced in the Eurozone’s smaller economies.

For politicians untroubled by scruple, this means the EU is an excellent excuse for failure – if in doubt, blame Brussels – but any Greek will tell you the constraints are real and lacerating. Mair argued that this structure of mutual frustration, which insulates great swathes of policy from popular democratic control, would eventually amount to more than dissatisfied grumblings about legitimacy. Instead politics would be increasingly conducted under the baleful light of Tocqueville’s question: if your political elite can’t achieve anything, or your vote doesn’t influence what they do, why should you put up with them at all?

It isn’t surprising that Eurosceptic populists have done well in these elections. It’s more notable that few of them now pursue explicit exit strategies, having observed Britain’s gradual unhinging. They prefer to highlight traditional European maladies: EU overreach and corruption, asylum and migration, fiscal headaches in the Eurozone. Euroscepticism has been a greater recruiting sergeant for the right than the left, with French voters far preferring Marine Le Pen’s sleekly repackaged operation to Jean-Luc Mélenchon’s avowedly eurosceptic France Insoumise. Left-wing eurosceptics generally understand such defeats to mean they need to be more dogged and severe in communicating Brussels’ antipathy to socialism, and more unwavering in their hostility; but it is hard to imagine how much harsher Mélenchon could be.

The French Greens, mutable on the left-right axis but enthusiastically European, were the surprise success. It may well be that increasing concern at the climate crisis has softened left-inclined voters, the need for international co-operation on climate change reconciling them to imperfect European institutions.

The UK results were unsurprising. The Brexit Party’s success was easy to predict; Labour took a slightly worse drubbing than expected; the Conservatives’ vote flatlined; the Liberal Democrats and Greens both profited. The prominent racist Stephen Yaxley-Lennon lost his deposit in North West England, and Ukip, having transformed itself into the political wing of YouTube incels, collapsed.

There will be a strong instinct in the core of both Labour and the Tories to look for reasons to discount the results: European elections are not good predictors of general elections – Labour lost half their MEPs in 1999 but held their majority in the Commons two years later – and the Brexit Party seems to have absorbed the Ukip vote from 2014; the Lib Dems, too, have a stronger record in European elections than elsewhere. But these excuses pale next to the headline figures: Labour’s result is its worst at the national level in a century (just slightly worse than Brown’s in 2009); the Tories’ the worst since their foundation as a political party.

Labour’s results confirm what began to be apparent at the local elections: as Brexit increasingly defines the political conversation, both ends of Labour’s electoral coalition begin to fray – they did badly in both Islington and Bolsover. Among hard Remainers, Michael Ashcroft’s polls show strong movement to both the Lib Dems and the Greens (39 per cent of 2017 Labour voters who took part in these elections); half of those who switched to the Lib Dems say they expect to stay with them for the next general election. There were similar though less pronounced movements among Leavers deserting Labour for the Brexit Party (13 per cent).

Insofar as Labour’s Brexit strategy was designed to reduce the governing party to a state of civil war and self-immolation, it has worked. But it has done so at the cost of losing its activists’ enthusiasm and voters’ trust. The toll has not been fatal (turnout among both activists and the general public was low), but it should be understood as a sign of a real problem.

If diagnosis is relatively easy, treatment is harder. The most enthusiastic pro-Europeans claim that only a full-throated conversion to Remain – whether by revoking Article 50 or campaigning for it in a popular vote – will fully recapture the young, educated and volatile end of the Labour coalition. They point out that both the party membership and Labour voters, even in pro-Leave areas, is predominantly pro-Remain. The leadership has been hesitant to embrace this strategy, partly because it rightly fears that the class politics it prizes will be subducted under – and destroyed by – a Brexit culture war. The Corbynite hope (evident in recent speeches) has been that speaking neither to the 48 per cent, nor to the 52 per cent, but to the 99 per cent, would allow British politics to repolarise along lines more amenable to Labour’s politics – that there ought to be a party capable of speaking to the vast majority of British society. The risk is looking like you’re talking to nobody at all.

Many of the Labour Party’s intrinsic problems have been brought to the fore in the Brexit debate: a mandarin conception of the relationship between party, membership and wider class, which resulted in the awkward drafting of the original Clause IV (‘to secure for the workers by hand or brain’) and the gradual isolation of the professional party from its rank-and-file membership; a willingness to treat certain parts of its electoral base as if they have ‘nowhere else to go’ (as one of Blair’s advisers once told his cabinet); a staid grasp of the composition of the working class; a difficulty in grasping constitutional questions as anything other than an ideological excrescence of underlying class relations (the results in Wales and Scotland ought to underline this). The party may have rediscovered socialism, but all these issues have yet to be tackled. It is tempting to see them as the major obstacles to ‘resolving’ Brexit – they have certainly all contributed to Labour’s occasionally obscure position on it.

Corbyn’s detractors, frustrated supporters and extramural Remain campaigners all accord him too much power. He has only a weak disciplinary grip on the PLP: there was a three-line whip to vote for a second referendum; front-benchers resigned in order to vote against it. Much of Labour’s Parliamentary strategy has been about avoiding splits, and refusing to give cover for MPs who are sympathetic to May’s deal to peel off and squeeze it through.

Agency now lies with May’s successor – who will almost certainly have to anoint himself the most frothing of No Dealers to get through the Tory selectorate – and the scope for Parliamentary action by the Labour Party is narrow. The new PM is unlikely to wish to prorogue Parliament and open a new session, providing an opportunity for tests of confidence and amendments to a Queen’s Speech. There could be a Labour-led confidence motion, but Change UK MPs are unlikely to support it, as it would mean torching their own seats. That closes off the only obvious routes to a general election.

Labour policy commits the party to a second referendum in that case, but it’s hard to see how they could bring it about if the government declines to introduce legislation, but instead tries to use the summer recess and party conference periods to wait out the Brexit clock. The Parliamentary innovations Labour has used in recent months – to force the production of documents, or alter standing orders – may again be needed.

Green bench guerrilla movements aside, the mood of the membership is essential to delivering on whatever position the party takes if another referendum is forthcoming. Enthused members can pull off extraordinary feats of campaigning, as they did in 2017; they’re unlikely to be so inspired for a second referendum, but they must be allowed to campaign on what they collectively believe, or have had a chance to decide. Otherwise de Tocqueville’s question – what are political elites for, anyway? – will return to haunt the Labour Party again, too.