Yes, it would be lovely to sit and sing Kumbaya, but when there is a situation in which there is no compromise - if one person attends the other can't - then it won't work, will it?
And however much you enjoy provoking people, you won't rattle me with your tit-for-tat insinuations that asking you this question is 'masculine', or that needing sensible rules is somehow perpetuating the patriarchy?.
People need to have a baseline - someone 'refusing to take a side', and being 'astounded' that they need to know their rights is no help whatsoever.
Obviously this was a hypothetical situation, but the bottom line is that you have been given another opportunity to say whether you would put a transwoman's wants above those of a woman's, and yet again you have done so. At best, the woman in your scenario has to give up half of her swimming hour, if she agrees to your compromise solution. The transwoman gains that half hour, as she is able to access the mixed sessions if she wishes.
You might argue that giving up swimming is not a big deal (obviously, I wouldn't), but it is far more serious for the woman who has to share a cell with a transwoman, or for the woman who has English as a second language and doesn't realise that she is a 'body with a vagina', or a 'person with a cervix', so doesn't get her smear test, in case a transwoman takes offence at the term 'Well-woman's clinic'.
These days, there is little time for doctors to see their patients at all, much less an opportunity for them to sit them down nicely and explain that they are no longer 'women', but 'people with cervices'.
Out of interest, who wrote the bible of 'real feminism'? I wasn't aware that there was a root of it, or that there was a vision we should all blindly follow - when did that happen?