There would be a marker on it to explain the mismatch.
Also, depending on the circumstances, the marker might or might not be relevant. It would always be for sport, of course.
We don't use the term intersex on these boards because we were specifically asked to refrain by a representative of an advocacy group for people living with DSDs.
However, across the Atlantic, "intersex" seems to be an accepted term to describe the tiny fraction of DSDs which result in ambiguous genitalia at birth.
Swyer's Syndrome must be an incredibly difficult condition to get your head around, especially if you only discover it at puberty. So you have my sympathy, OP.
Whether or not someone with a cross-phenotype (I've just made that term up!) is deemed the opposite sex depends on the context. I'm not talking about the sex they choose to present as, I mean what category they fit best in when it comes to hospital wards, sporting categories, the Oscars, single sex services, etc. All should be delicately handled, but the other people in the category have rights, needs, etc as well. That should not be forgotten.
I sometimes wonder if it would be better for us all if DSDs were more widely known about and understood - but I am aware that it's not my issue.