Jeremy Corbyn has reiterated his pledge in the Labour manifesto to allow people to self-identify their gender. He told a Commons reception: "Discrimination has gone on too long. The Gender Recognition Act does not allow trans people to self-identify their gender and forces them to undergo invasive medical tests. This is wrong."

Theresa May has already said that she wants "changes to be made" to the current system, but the feeling in Westminster is that the socially conservative DUP is unlikely to back her. Hence Corbyn's offer.

Now, the language Corbyn has used here doesn't fill me with confidence that he understands the current system for gender recognition. Currently, a gender recognition certificate is issued by a special panel to an individual after two years living in their acquired gender. There is no requirement for what used to be called "sex change surgery", although panels do ask for medical records showing a diagnosis of gender dysphoria. Applicants are also asked to show other IDs in their new name. But Corbyn's phrase "invasive medical tests" seems to invoke the idea of punitive and humiliating procedures like the attempts in some US states to give mandatory vaginal ultrasounds for anyone wanting an abortion. In the UK, if anyone is looking at your genitals, it's for medical rather than legal reasons.

The new self-ID system was proposed by the Commons Women and Equalities committee, chaired by the former Conservative cabinet minister Maria Miller. It's a bit dismaying that this is the only bit of the committee's recommendations which has achieved any traction, because the report had some fairly stern words about the misery caused to trans people through their difficulty in accessing specialist NHS services. But, as I wrote at the time, fixing the lack of NHS funding is a lot more expensive than changing the GRC system. (Incidentally, Miller said at the time of the report being released that the only opposition came from women "purporting to be feminists"; I am in no way surprised by this. I doubt if I stopped the average person in Britain - or even the average male MP in Westminster - they would have no strong opinions on the potential drawbacks of the self-ID system. Feminists are literally the only group dedicated to interrogating whether proposed policies are bad for women. It's not that everyone else necessarily thinks her proposal was a good idea; it's more that they haven't thought about it at all.)

There are undoubtedly drawbacks to the current system. There is a fee, although a means-tested refund is available. It is drawn-out and bureaucratic. As one trans woman I talked to earlier today pointed out, trying to navigate government bureaucracy at a difficult and disruptive time of your life is no one's idea of fun. That perhaps explains why some of the trans women who have come to public attention for being housed in male prisons didn't have a certificate. As with any tangle with government agencies, the GRC process is a lot easier to navigate for sharp-elbowed, well-educated, well-spoken people with English as their first language.

But there are two critiques of self-ID. The first is from the transgender perspective: that it will, put bluntly, augment the ranks of "transsexuals" (not the best word, but you will see why I've used it in a minute) with the much wider category of transgender people. The cutting edge activist understanding of gender is one that even some older trans people find fairly radical: that you can be trans without any dysphoria (hatred of/alienation from your sexed body) and that gender is purely about an inner essence. On Tumblr, those who follow the older strand of thinking - that being trans involves living as the other sex - are sometimes referred to as "truscum" or HBJers, after the sexologist Harry Benjamin, who developed protocols for the treatment of trans people.

This divide doesn't get talked about much, for the simple reason that one of the repeated criticisms thrown at trans people is that they are "pretending" or "putting it on". So those with a dysphoria diagnosis are wary of implying that those without one are "not really trans". But my sense is that to the general population, there is a difference between somebody who has physically transitioned and someone who, say, identifies as non-binary genderqueer but hasn't pursued a transition of any sort. There are even some trans women who worry acutely that cross-dressers, particularly those who do so for fetish reasons, reduce support for their right to live as women by seeming to parody ideas of femininity while appearing very male. Other activists are concerned that the currently very low rates of detransition could climb if there is no gatekeeping - again, potentially giving fuel to those who see this as a trend or fad.

The other critique of self-ID is the feminist one. It runs broadly like this: sex-segregation is now limited in public life, and it is done primarily for reasons of safety or fair competition. (Think: domestic violence shelters, and competitive sports.) The Equalities Act 2010 has "gender reassignment" as a protected characteristic, so that discrimination against anyone on those grounds is illegal. There are very limited exceptions, which a move to "gender identity" as the characteristic could remove.

For some reason, this debate often ends up talking about toilets. That's probably because online activism in English is dominated by the US, and in America, a series of punitive “bathroom bills” is being pushed by conservative evangelicals, ostensibly to protect women, but really to make life harder for transgender people. (The bills would require people to use the toilet which corresponds to their birth sex.) There is no legislative parallel in Britain because of the gender recognition certificate - once you get that, you get a birth certificate which shows an "M" or "F" just the same as if that had been your sex from birth. Also, we are British and probably wouldn't stand for our police hustling into loos and demanding to see someone's driver's licence.

For feminists, it's the whole idea of legislating on "identity" rather than material conditions which is potentially controversial.

For some characteristics, we already use pure self-declaration (like race) on the understanding that public services do not rely on racial segregation. However, there are still some areas of life which do operate sex-segregation, from rape shelters to professional sports to women’s swimming lessons to hospital wards. Sometimes, the segregation is for safety; at other times, it encourages marginalised religious communities to use facilities which women might otherwise be forbidden from visiting (like gyms or swimming pools). In sport, it is done because every credible scientist acknowledges that testosterone and male puberty confer huge advantages in many sports.

Much has been made of the plight of trans women in prisons. In the UK the department of justice has tried to steer a course which protects both the principle of sex-segregation and trans women’s sense of self. (There is generally acknowledged to be no problem with trans men who want to move to the men's estate.) The presumption is that trans prisoners should live in the estate of their professed gender, if they are living in that gender, unless a case review board decides otherwise. Edward/Joanne Latham, who was found hanged, is often rolled into an aggregate number of trans prisoners who died by suicide, but the inquest heard that Latham “went through phases”. Staff addressed her as Joanne, but she had legally changed her name to Edward earlier that year. (Latham was also highly unstable and violent, unlike the vast majority of inmates in the female estate). A counterpoint is the case of Paris Green, who murdered a man alongside two friends, after sexually assaulting him with a rolling pin. She had to be twice moved from the women’s estate after having sex with female inmates.

It should be noted here that the prison service says that the vast majority of trans prisoners are accommodated in the estate they feel best suits their gender identity. Our prisons are brutal, understaffed and underfunded, with terrible provision for those with mental health difficulties or addictions, and a horrifying number of people kill themselves every year. Trans women are obvious targets for bullying and sexual assault. But that does not negate the fact that sex-segregation of prison is necessary for women’s safety. A move to pure self-declaration is obviously open to abuse.

I use prisons as an example of competing rights as it is perhaps the most difficult example because of the demographics involved. But similar concerns apply to women’s refuges. Again, the sector must balance trying to support trans women (who have access to little or no specialist services) with their other service users. And the Olympic committee and others have wrestled for years with the fairest way to draw a boundary around competitive sports, given the potential benefits of high testosterone to athletic performance. Recently, women with naturally high testerosterone were barred from competitions unless they took medication to lower it, although that has now been overturned. But clearly, the blurry boundaries of biological sex (never mind cultural gender) are going to present the same questions to sporting authorities as other physical advantages, such as those who have a naturally higher number of red blood cells. Feminists are worried that the move to self-ID will demolish the rationale for, and the operation of, single-sex sports.

As I've just demonstrated at exhausting length, this proposed change is a quietly radical one. It has potentially big repercussions for everything from the justice system to the actuarial life-expectancy tables used to calculate pension pots. It might suit politicians like Maria Miller and Jeremy Corbyn to present it as something very straightforward, but it isn't. If those politicians really respect their voters, they will come clean about the competing rights involved.