The conflation between what the law is, what gender critical people want, and want they want the law to say is quite remarkable. It's all set out in the EqA.
You can have single-sex spaces. These SSS include people with the protected characteristic of gender reassignment, which is "the person is proposing to undergo, is undergoing or has undergone a process (or part of a process) for the purpose of reassigning the person's sex by changing physiological or other attributes of sex" by default.
Note the use of the word "sex."
You can exclude transgender people from single-sex spaces which match their acquired sex under paragraph 28. There must be an objective reason.
28(1)A person does not contravene section 29, so far as relating to gender reassignment discrimination, only because of anything done in relation to a matter within sub-paragraph (2) if the conduct in question is a proportionate means of achieving a legitimate aim.
(2)The matters are—
(a)the provision of separate services for persons of each sex;
(b)the provision of separate services differently for persons of each sex;
(c)the provision of a service only to persons of one sex.
The current statutory code clarifies this:
If a service provider provides single or separate sex services for women and men, or provides services differently to women and men, they should treat transsexual people according to the gender role in which they present. However, the Act does permit the service provider to provide a different service or exclude a person from the service who is proposing to undergo, is undergoing or who has undergone gender reassignment. This will only be lawful where the exclusion is a proportionate means of achieving a legitimate aim
When the AEA challnged this, they lost - they were found "wrong in law."
From AEA vs EHRC.
"The claimant submits that if a difference of treatment can be justified vis-a-vis birth men in general, then it is inconceivable that it cannot equally be justified vis-Ã -vis birth men who are transsexual women. On that approach, though, the Equality Act's gender reassignment provisions would in substance provide no protection at all, in the context of an SSS, to transexual persons without a GRC. ... So if vis-Ã -vis men in general it is a proportionate means of achieving a legitimate end, then the same must apply vis-Ã -vis birth males who are transexual women. Thus, the claimant's approach would place transsexual women without a GRC in the same position for these purposes as all other birth males. That is clearly incompatible with the tenor of the Act.
"In deciding whether a PCP is a proportionate way of achieving a legitimate end, it is inevitable that regard must be had to its impact on persons with the protected characteristic in question. It is clearly wrong to assume, as a matter of law, or as a matter of obvious practice, that the answer will necessarily be the same whether one assesses a PCP as applied to birth males in general or whether one assesses it as applied vis-Ã -vis birth males who are transsexual women."
However, it is in my view clear beyond argument that Parliament has chosen, in the 2010 Act, to place transsexual persons in a different position from the generality of persons of their birth sex. What effect that has in particular
circumstances will depend upon the application of the provisions in the Act, including the Schedule 3, para. 28 justification exception, to those circumstances. I do not accept the claimant's contention that the Code makes clear errors of law in the way in which it sets the position out. On the contrary, I consider the claimant's construction of the Act itself to be clearly wrong in law for the reasons I have summarised.