A comment at your link baldly asserts that ZF+CH implies choice for sets of real numbers, but the link seems otherwise irrelevant. Do you have a better citation? In particular, what do you mean by CH without choice? In fact, the comment asserts that ZF+CH implies R is well-orderable, which I don't think is true under weaker notions of CH.
In particular, what do you mean by CH without choice?
CH in that context then is just that there are no sets of cardinality between that of R and N. You can't phrase it in terms of alephs (since without choice alephs aren't necessarily well-defined). As for a citation, I think Caicedo's argument here can be adopted to prove the statement in question.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.