The elections that were not supposed to happen have come and gone, leaving behind a shattered two-party system.

The Conservative Party has recorded its worst ever result in a national election, coming in fifth with a meagre 9 per cent of the vote. The Labour Party fared better, but not by much, losing half of its MEPs and recording its lowest nationwide popular vote in 100 years, or its lowest ever since the introduction of universal suffrage. The newly formed Brexit Party came in first, both in seats and vote share, as I’m sure you’ve heard. While it performed only marginally better than UKIP in 2014, it remains a truly impressive achievement for a party which only came into existence some three months ago. The Liberal Democrats have further established themselves as the biggest anti-Brexit force in British politics, beating Labour into third place and humiliating it in its London heartland. The Greens achieved their best ever result in a European election, or any UK wide election for that matter, winning seven seats and over two million votes.

Looking beyond the headlines, what can these results tell us about attitudes towards Brexit? Both sides of the divide should probably curb their enthusiasm. On the one hand, it is true that this was a largely single-issue election. A Lord Ashcroft poll conducted after the vote found that Brexit Party, Lib Dem and Green voters were chiefly motivated by Brexit, while those that stuck with the two main parties did so mainly out of tribalism. However, we probably shouldn’t treat these elections as a de facto second referendum, given that we cannot be certain where the 20 or so per cent of those who voted Labour and Conservative lie, and the same could be said for nationalist parties in a Scotland, Wales and Northern Ireland. No matter how tempting it might be to add up vote shares of the various parties in order to prove a point, I would ultimately advise against such an exercise (at least if you happen to value honesty over partisan point scoring).

If there is a story to be told here, it is one of increasing polarisation and enduring national gridlock. Those on the extremes of the Brexit debate prosper, and those in the middle find themselves rapidly abandoned by voters. Meanwhile, without participating in some rather intense mental gymnastics, it is difficult to find a majority for anything. While the Brexit Party performed very well, it still secured under a third of the popular vote. Combine their support with whatever is left of UKIP, and it is clear that Thursday’s vote did not produce anything resembling a majority for no deal. The same can be said for non-nationalist parties explicitly in favour of a confirmatory referendum.

A secondary question is what, if anything, this tells us about future electoral prospects. In my breakdown of the local election results, I cautioned against using second-order contests to predict what might happen in a general election. Such a warning is only more pertinent here, considering the dominance of Brexit throughout the campaign and the fact that these elections were conducted under an entirely unique (to the UK) electoral system. We simply cannot expect voters to behave the same when other issues are at play and the stakes are entirely different. Certainly, when faced with the prospect of a Corbyn government, we can expect far fewer Conservatives to lend Nigel Farage a protest vote. Similarly, do not expect quite as many Labour remainers to desert when the consequence could be five more years of Tory rule. The Lib Dem surge is not quite about to eject Jeremy Corbyn from Islington.

Once again, it is a far better use of our time to observe wider trends as opposed to applying nationwide swings to entirely different contests (as a side note, if you input these European election results into Electoral Calculus, it essentially breaks and gives the Brexit Party an enormous majority. While the odds of that actually happening at a future general election are close to zero, the fact that any party could theoretically win over 400 seats with under a third of the vote is a rather grave incitement on our electoral system). The wider trend in question here is essentially a confirmation of what I wrote after the local elections – “leave voters can no longer be counted on to save the Tory party, and nor will remainers go out of their way for Labour”.

While we can expect both parties to keep their existing coalitions far more intact in a general election, you don’t need half of either Labour or Conservative voters to defect in order for the other side to win. With the next election likely to be decided by a few dozen key seats, it might only take a modest Brexit Party spoiler effect in places like Swindon and Stoke South for a Labour government to become inevitable. It is becoming increasingly clear that the resurgence of two-party politics in 2017 was all but an anomaly, and the next general election will almost certainly see both main parties losing a considerable amount of vote share. The big question now, and which will no doubt occupy the minds of party strategists, is whether Labour can lose fewer remainers than the Conservatives will inevitably shed leavers, and vice versa.

This also brings us to how this election will impact the internal struggles within Labour and the Conservatives alike. As far as the former is concerned, it has become impossible to deny that Labour’s strategy of trying to appeal to both sides of the Brexit divide, simultaneously wanting to leave the European Union but attempting to keep remainers on side through vague promises of a “public vote” if it cannot achieve the kind of deal it wants, has failed spectacularly. The party’s leaders have admitted as much on Monday. Labour’s Brexit policy will have to change, but how exactly might this be done remains an open question.

On the one hand, the answer should be simple: accept that a snap general election won’t happen anytime soon, embrace remain and campaign unambiguously for a confirmatory referendum on whichever Brexit option the next Conservative leader adopts. This really should not be a difficult choice to make. The majority of Labour MPs and the overwhelming majority of its members and voters oppose Brexit. The party lost four times as many voters to the Lib Dems and Greens than it lost to the Brexit Party.

So what is holding the Labour back? Its leadership remains dominated by those, like Jeremy Corbyn, who never had much love for the European project, and a fresh referendum continues to be opposed by its most influential trade union backers (chiefly Len McClusky of Unite). Meanwhile, a significant cohort of MPs, which includes Lisa Nandy, Caroline Flint and Stephen Kinnock, oppose a confirmatory referendum on the basis that it will only further alienate Labour’s traditional working-class base. But such concerns fundamentally misunderstand what Labour needs to do in order to win the next general election.

Since 2015, the party has lost huge chunks of its older working-class supporters and replaced them with younger metropolitan liberals. Winning back the former might be a worthy enough endeavour, but it will prove fatally counterproductive if it is done in such a way that alienates the latter. Furthermore, while a majority of Labour seats did vote to leave the European Union, the majority of Labour voters in those seats did not. As mentioned above, Labour does not really need to expand its coalition in order to win, it just needs to keep it more intact than the Conservatives can hold together theirs. Even in solid leave seats, the party can still come ahead on the back of remain voters as long as leave voters are sufficiently split between the Tory and Brexit parties. First past the post is a powerful drug.

So what of the Conservatives? Weeks away from a full-blown leadership contest, the outcome of the European elections will no doubt confirm in the minds of Tory MPs and members that their party must deliver Brexit. Unlike Labour, I expect it will make the electorally obvious decision very soon, and support the policy backed by the overwhelming proportion of its voters (deal or no deal, Brexit must happen). Unfortunately for them, as a united and single-issue driven force, the Brexit Party presents an arguably greater threat to the Conservatives than the various anti-Brexit parties do to Labour. They also face an attack on two fronts, with a Liberal Democrat revival in the South West threatening a number of key seats won at the 2015 general election. Whoever ends up being the next Conservative leader certainly has their work cut out.

Finally, we should not forget that last week’s European elections were, well, European. Nearly 200 million people voted across the continent, with the highest recorded turnout 25 years, and the highest ever following the bloc’s 2004 expansion. In Poland, turnout increased by over 20 per cent. In Romania, by 16 per cent. For anyone who wants the European Parliament to be seen as a legitimate institution, that alone is welcome news.

As for the results themselves, the main story is that of fragmentation, with the centre-right and centre-left blocs that have long dominated in the European Parliament losing their majority to a range of Liberal, Green and populist parties. Generally speaking, the much prophesied populist rebellion has failed to materialise. While both the ENF (right-wing nationalist) and EFD (Eurosceptic) blocs made gains, their MEPs still hold a combined 112, or about 15 per cent, of the total 751 seats in the European Parliament, fewer than the Liberal and Green blocs and fewer still than the embattled Conservatives and Social Democrats. While populist gains were limited outside of Italy, where the Lega won a resounding victory, their presence in Brussels has been entrenched nevertheless.