This is a story about how politics fails. It starts with a hairline crack that slowly widens, until it’s big enough for some people to slip through. Eventually, the gap becomes a chasm. And if nobody builds a bridge, eventually the other side almost disappears from sight. This week brought another small earthquake along the faultline running through the Labour party over trans rights. Three women running for the leadership – Rebecca Long-Bailey, Lisa Nandy and Emily Thornberry – all signed a pledge drawn up by a hitherto unknown group of trans activists demanding a battle against what it calls transphobic “hate groups”. (The lone male candidate, Keir Starmer, didn’t sign; Thornberry warned against using the phrase “hate group”.) The pledge specifically named Woman’s Place UK (WPUK), a grouping of leftwing feminists and trade unionists who insist they don’t hate anybody but do worry about trans women accessing all-female spaces – such as domestic violence refuges, prisons, changing rooms and toilets, given proposed reforms making it easier to transition legally. The hashtag launched by defiant supporters of Woman’s Place UK – #ExpelMe, daring Labour to either kick them out or have the guts to defend them, a choice the party seems desperate to avoid – reflects long-simmering tensions.

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This isn’t just another factional fight pitting left against right or millennial against boomer. Even Laura Pidcock, a 32-year-old Corbynite rising star who recently lost her seat, was loudly criticised online for writing that “the women’s movement needs the space to talk about sex and gender without fear of being no-platformed”. It’s difficult to tell how many MPs agree when so many now nervously swerve the whole topic. But there’s nowhere to hide in a leadership contest. When Nandy told a recent hustings that “trans women are women” there were whoops of delight. But ask around for women feeling alienated from Labour over this issue and they’re easily found.

Some are survivors of domestic violence who resent being told to “educate yourself”. Others are bewildered that decades of loyally knocking on doors seemingly hasn’t earned them the right to be heard. They long to hear someone defend what is still Labour’s official policy: promoting trans equality but defending powers in equality law that let organisations exclude trans women from all-female spaces in exceptional circumstances. Yet even Starmer goes no further than urging people to “dial it down”.

Why is the left, home of liberation struggles, making such hard work of this?

Why is the left, home of liberation struggles, making such hard work of this? It was a Tory government that fumbled trans rights reform, rushing out plans for people to self-identify but running away when that got controversial. Yet it’s Labour and the Liberal Democrats ripping themselves apart over it. Where downtrodden minorities revolt against oppressive majorities, progressives know which side they’re on, but this is their Sophie’s Choice: two marginalised groups, each demanding protection from the other.

Identity politics, rooted as it is in individuals’ feelings about themselves, is perhaps uniquely challenging for the left. The Tories are comfortably a party of individuals but Labour is one of collective struggle, and now solidarity is breaking down. Women who have waved placards all their lives now find themselves picketed, outside WPUK meetings, by people they’d once have marched alongside. Tensions between some trans and lesbian activists strain LGBT alliances (the tale of how this pledge emerged is People’s Front of Judea stuff, but suffice to say there are now two rival trans-friendly charters circulating, with Starmer backing one from the long-established LGBT+ Labour grouping). If the next leader’s task is bringing Labour together, this doesn’t bode well.

Yet it’s still not too late to find common ground. No compassionate human being should want a woman who has been raped or brutalised to feel traumatised all over again by sharing counselling or refuge services with someone they perceive as a threat. Even a person who poses no danger whatsoever can inadvertently frighten a traumatised person, if something about them – a sound, a scent, a habit – triggers flashbacks. But nobody should want trans people to feel unsafe or cast out, and barring a trans woman from women’s services seems the cruellest of personal repudiations. That much is common ground, and beyond the boiling hate of social media, people are building on it.

Some refuges have now accepted trans women (excellent risk assessment helps, and careful laying out of accommodation). Some schools absorb gender-questioning pupils without fuss; teens queue happily for mixed Topshop changing rooms; and with time, maybe we’ll wonder why unisex loos were ever an issue.

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But solutions don’t come from denying there is something to solve, and that’s why any future leader should have paused over point five of the Labour Campaign for Trans Rights charter, requiring candidates to accept there are “no material conflicts” between cis women’s rights and those of trans women.

For what some women hear when they hear those words is that there’s conflict, all right, but not legitimate conflict; no honest reason to get upset, beyond bigotry or ignorance, and therefore nothing substantial to discuss. As Pidcock discovered, the next step is outrage at the very idea of women still wanting to discuss it. Conversation closed.

To acknowledge that two sides of an argument exist is not to agree with every aspect of either. But the road to change is littered with objections, and the point of politics is not to ban them: it’s to work patiently through the serious ones while always moving forward. Labour needs a leader who can talk as well as walk.

• Gaby Hinsliff is a Guardian columnist