One thing that's been hard to keep up with, given our own (US) flood of political insanity: it seems like the UK is moving in an anti-transgender direction even more decisively than we are? Do you have any thoughts or clarification on that topic?
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This depends on your point of view. The Equality Act 2010 protects people based on nine specific characteristics: age, disability, gender reassignment, marriage and civil partnership, pregnancy and maternity, race, religion or belief, sex, and sexual orientation. The Act covers more or less everything situation you can think of where it matters -- want to rent out your spare room but the applicant is gay ? You would fall foul of the law. Sack a women just before she goes on maternity purely to avoid paying her leave, you will be in hot water. Refuse to serve someone in your restaurant who has changed gender - then see you in court.
The problem occurred between the two protected characteristics sex and gender reassignment. Some people believed that if you wanted to set up a space exclusively for people
without a Y chromosome to for example support them if they have been raped, then that was fine as sex is a protected characteristic and gender reassignment is not the same thing as sex. Some people believed otherwise.
The Supreme Court has just ruled that sex means sex - not gender. The way the act is written means any other interpretation would make the law unworkable. It might be the case the equality act is amended afterwards, who knows what the future might bring. When polled, the public tends to lean towards sex means sex, not gender and keeping for example, a rapist out of a woman's prison is the right thing to do even if they have changed gender. A very significant minority believe that is transphobic and just because someone has forced themselves on a woman, it would be wrong for other women to complain about being housed in close quarters with such a person regardless of their physicality.
The fall out of the law clarification is many-fold, companies and institutions who believed that transgender women are women regardless of the steps of transition (i.e. if a person declares they are a woman, but makes no changes at all to their clothes, appearance let alone hormones and surgery - so that they appear just the same as before they transitioned) have rapidly changed their polices to keep transgender women out of spaces that were for designated for women only.
Stonewall had made a killing out of advising companies on the law, but the Supreme Courts ruling has made all their expensive advise wrong. They are now in serious financial trouble because of this and other blunders in court. This reduction of funding of Stonewall has been touted as transphobic too.
There is still much to unravel, in large part as companies and government institutions trusted Stonewall to interpret the law. Transmen are going to be in a bad place until matters are more settled.
In short -- the public at large is not anti-transgender in day to day life, they would not support actions by groups or companies to deny them for example access to renting a home, getting a job, eating at a restaurant, getting justice for being physically attacked. They do not think that support services designed for women should have to support transwomen but would have no issue at all in a support service that supports transwomen explicitly and excludes women.
There is a lot more to the whole saga, including the primary clinic that looked after transgender children was found to be appallingly bad at their job. Hormones were given out too quickly and without examination if other factors were involved (autism in girls (XX) was found to be far too common to be ignored - but was), patient follow ups were not made, success/failure was records were not created or studied to examine for efficacy of their work. Some believed this shutting down was part of the anti-trans movement.