The statutory code says:
2.27
Transsexual people should not be routinely asked to produce their Gender Recognition Certificate as evidence of their legal gender. Such a request would compromise a transsexual person’s right to privacy. If a service provider requires proof of a person’s legal gender, then their (new) birth certificate should be sufficient confirmation.
13.59
Service providers should be aware that where a transsexual person is visually and for all practical purposes indistinguishable from a non-transsexual person of that gender, they should normally be treated according to their acquired gender, unless there are strong reasons to the contrary.
Statutory code is not quite the law but carries a lot of legal weight in courts and tribunals.
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The technical guidance says:
Where a transsexual person is visually and for all practical purposes indistinguishable from someone of their preferred gender, they should normally be treated according to their acquired gender unless there are strong reasons not to do so
Where someone has a gender recognition certificate they should be treated in their acquired gender for all purposes and therefore should not be excluded from single sex services.
Technical guidance carries less weight than statutory code but organisations may still be asked to justify why they have not followed it.
IMO the second paragraph here goes much further than the EA itself. The example in the EA explanatory notes (Schedule 3, part 7, para 28) is:
A group counselling session is provided for female victims of sexual assault. The organisers do not allow transsexual people to attend as they judge that the clients who attend the group session are unlikely to do so if a male-to-female transsexual person was also there. This would be lawful.
Would the clients be suddenly willing to attend if the trans person had a GRC?
(by the way, I couldn't get the PDFs on these pages to download directly. Right click - copy link location - paste into new tab works for me)