How do you solve a problem like the Green Party? Their surge in support is giving all the parties headaches of one kind or another.

Vince Cable thinks he knows the answer: love-bombing. At the 2017 general election the Liberal Democrat leader helped to broker a pact with Caroline Lucas to prevent the two parties standing against each other in several seats. These days, he likes to talk up the Lib Dems’ love of environmentalism to defuse the Green threat, rather than challenging his rivals directly. He knows both parties are competing for Remain voters.

That loving feeling is not entirely mutual. Caroline Lucas has now been succeeded as Green leader by co-leaders Siân Berry and Jonathan Bartley. The former likes to remind voters of the Lib Dems’ participation in the 2010 coalition with the Conservatives. That irritates some Lib Dem MPs, who worry that Cable is too friendly with a party with which he is competing for votes. Then again, Cable has earned some latitude. He is basking in the success of the party’s local election campaign, which brought a net gain of 706 council seats. The Lib Dems also expect to do well in the European elections on 23 May.

Cable’s strategy has another point in its favour. Cross-party co-operation is a Liberal Democrat value. Its grass roots are probably the only activists in British politics for whom the pledge to stand aside in favour of nominal opponents is an applause line.

Labour also prefers to talk up its own environmental credentials than engage in gloves-off combat with the Greens. However, the rise of the Greens is a source of greater anxiety in Britain’s main opposition party, not least because Labour’s internal culture is less welcoming to cross-party alliances and compromise.

The success of the Greens in the polls – and more importantly in the local elections – raises questions that Labour cannot easily answer. For months, the party has faced questions over Jeremy Corbyn’s poor personal ratings and its underwhelming electoral performance. (It lost 84 councillors overall in the local elections.) Until now, the ready answer among Corbynite loyalists has been to say that outside election periods the broadcasters have treated Corbyn unfairly, which has depressed his vote share. They argue that Labour’s electoral coalition of the young, ethnic minorities of all ages and people who live in big cities is inclined not to show up for second-order elections, such as the local or European contests.

These are impeccable arguments to explain Labour’s underperformance in comparison with the Conservative Party, which has a solid core of voters who turn out reliably, and which benefits from a favourable media ecosystem. However, such arguments dissolve upon contact with the Green Party. The Greens struggle to receive any coverage, let alone fair coverage. That frustrates party activists, who feel that the past three years have shown that the best way to get on the BBC is publicly to denigrate the BBC. But it is not a game they want to play, for fear of becoming “useful idiots”, in the words of one, for right-wingers who want to dismantle the corporation.

There is no flattering explanation for why the Greens – whose electoral coalition is also young people, ethnic minorities and big city dwellers – have been able to get out their vote when Labour hasn’t. The success of the Green Party, then, adds to a wider sense of unease in Labour. Everyone in Westminster believes that voters feel hostile towards the establishment. So why can’t Labour harness that feeling?

Since the Green Party is more radical than Labour on several major issues – climate change, drug reform and Brexit – MPs for the latter are wondering if their policies are the problem. Labour is undeniably timid on the decriminalisation of drugs – the issue is well-aired within the party – but no one thinks it is a vote-winning issue. On climate change, Labour has lots to say about investing in research that might create the environmentally friendly industries of the future, but much less on the industries causing ecological damage in the here and now. That leaves Brexit, which most MPs believe is doing the party damage. “Pro-Europeans want to vote for a pro-European party,” one MP told me. “In other news, the Pope is Catholic.”

Yet the evidence is not cut and dried. Labour asked the pollster Survation to investigate its local election results, and a copy of the findings was sent to every Labour MP. Survation did not believe that Brexit could explain the party’s failure to seize the anti-establishment mood. Where Labour did well, it did so in both Remain and Leave strongholds, while likewise underperforming in other regions across both sides of the referendum divide. These findings are supported by similar work by the respected psephologist John Curtice.

One group of Corbynites has a theory that Brexit is the problem, just not in the way you might think. The trouble with Corbyn’s Brexit policy is that it is transactional, designed to appease the party’s various power brokers and the demands of winning the next election. In other words, it is exactly the sort of artful triangulation that the Labour leader is meant to reject. This view gains added credence from the fact focus groups increasingly see him as just another politician.

Yet there is one crucial thing to remember. When it comes to the next general election, a surge in support for small parties would hurt both Conservatives and Labour in terms of raw voter numbers. But polls suggest those voters are geographically distributed in a way that means Labour would gain seats. That’s not the way that Jeremy Corbyn would want to become prime minister. But would that matter, when Downing Street was within his reach?