Before the local elections last week, the Conservative Party had said that losing a thousand councillors would be a disaster. In the event, the collapse of the Tory vote was more than three hundred seats worse than that. The wipeout in Chelmsford left the Tory MP, Vicky Ford, in tears; at a gathering of Welsh Conservatives, the prime minister was greeted with active heckling, a rare choice for the Tory grassroots, who generally prefer to dissent in truculent silence. Andrew Mitchell, a former chief whip, was ‘surprised anyone was bothered to vote for us’. At the coming European elections, with the Brexit Party in contention, the faithful remnant may be yet further diminished.

Tory failure did not translate into Labour gains: the Greens and Liberal Democrats were the ones smiling the morning after, alongside a host of microparties and independent associations. Average turnout was around 30 per cent. It’s hard to make observations about national politics from local elections: Labour’s poor showing in the locals two years ago implied a 27 per cent share of the popular vote; in the general election a month later it got more than 40 per cent. The traditional claim is that local elections allow voters to register their discontent with the main parties without the risk of dislodging an administration or undermining its main challenger. The argument after these elections isn’t whether such a protest took place, but over its political interpretation, with both sides of the Brexit divide claiming that it endorses their preferred outcome.

John McDonnell, the shadow chancellor, had hoped Labour would gain a few hundred seats; it ended up with a net loss of 63, and ceded control of six councils: a defeat of a different order of magnitude from the Conservatives’, though some of the losses – Hartlepool, Bolsover, Wirral – have symbolic weight. Sporadic good news for the party in target seats doesn’t make the search for explanations any easier. Most turn on two factors – base and Brexit – and neither offers much consolation. The Labour electoral base skews young, less advantaged and less securely housed, and is harder to turn out, especially without the high stakes of a national race. But one reason for Corbynism’s success has been its ability to draw unexpected or dormant constituencies to the polls: not this time.

Labour’s strategically ambiguous Brexit policy pleases no one, including the party leadership. Those sympathetic to Leave positions point out that Labour’s losses were concentrated in Leave-voting areas. Those sympathetic to Remain cite the explosive growth of pro-Remain parties, and argue that Labour voters, even in Leave areas, are predominantly against Brexit. Labour strategists have assumed that Leave voters are the more volatile end of their coalition, liable either to stay home or seek out a party less lukewarm about Brexit: perhaps this election shows they were wrong to count on Remainers’ votes.

Base and Brexit are both necessary but not sufficient explanations for Labour’s poor performance. Corbynism has most often expressed itself, on the national level, as an insurgent force intent on remaking politics: first the Labour Party, with its vastly increased membership and changed political horizons; next the country, intending (in John McDonnell’s words) an ‘irreversible shift in wealth and power’. It has emphasised transformation led from Westminster, and on a national scale.

Local government, still the strongest redoubt of the party’s right, has been mostly absent from that conversation. Administering budgets that have been slashed by successive rounds of cuts, incumbent Labour councils already faced hostile voters: it’s hard to combine insurgency with long-established dominance of the local administration; harder still when observing the general Labour omertà on any criticism of the party’s record. It is noteworthy, then, that in Preston, a Leave-voting city but also the site of the party’s most significant experiment in municipal socialism, Labour increased its majority on the council.

The response of both Theresa May and Jeremy Corbyn to the results – that they demonstrated a need to ‘deliver’ Brexit – seemed to many a ludicrous travesty of the actual vote. The Labour Party has little incentive to agree a Brexit deal with May: it will provoke open rebellion in the Parliamentary party, repel most of the activist base, and draw Labour into the Tories’ intensifying civil war as defenders of the prime minister. All this makes it unlikely that any deal will be reached. So why are they insisting on their commitment to reaching one?

For both parties, a lot turns on whether Brexit presages a deeper and longer-lasting political realignment, or the parties’ basic electoral coalitions will endure through to the other side of the Article 50 process. Much of the Conservatives’ base is at risk of flaking off to new parties braying loudly of Brexit ‘betrayal’. Labour is not only in danger of losing its anti-systemic energy to Brexit-related insurgents, but faces the difficulty of pushing a socialist politics in a national conversation dominated by arguments over membership of a liberal trading bloc.

Anti-systemic energies are difficult to contain once unleashed. Ballot-spoiling has usually been a minority pursuit in Britain, but substantial numbers – 800 in Basildon, 693 in Great Yarmouth – rejected the options on the ballot paper, with many demanding ‘deliver Brexit’, ‘May out’ or ‘traitor’. These are the preliminary squalls of the forthcoming European elections, and Nigel Farage is ready to capitalise on the discontent. The Brexit Party’s rallies have so far concentrated and amplified this anti-political mood, combining personal loathing of the prime minister with emotional appeals to a sense of betrayal. It is a potent message. The Remainer parties may have been the winners in this last election; it is hard to see how they will beat back the hardest of Leavers in the next one.