True story, did you know you can have paradoxical sets without choice? You just need infinity.

https://en.wikipedia.org/wiki/Paradoxical_set

http://www.math.hmc.edu/~su/papers.dir/banachtarski.pdf

While I don't dispute your claim, neither of the links substantiates it. The Wikipedia article says that you only need the axiom of infinity, but mentions only Banach–Tarski (which does require choice); and the minor thesis says on p. 2 that "the philosophy adopted in ths paper will be the unquestioned acceptance of Choice as a useful foundation in our work".

Do you have any other references for this? (I'm interested, not snarking.)

The paper is kind of long. Look at Theorem 4 in it.

https://www.math.hmc.edu/funfacts/ffiles/30001.1-2-8.shtml