I approve that but I would say that Street's paper was "not fully correct" instead of "wrong" :) As you said above, his paper still contains fundamental tools and ideas, which were necessary for me to find a fix, so that "wrong" feels to much discarding…

@TimCampion: Sorry about that … My article was already too long, so that I had to put a stop to details and remarks.

(Timer for editing, sorry) No, assuming the tightness of $\mu(x)$ is sufficient. As I said in my reply below, Theorem 4.2 does not hold fully with or without the corrigenda. Concerning the fact I made tightness part of the definition of Street, it seemed acceptable, since for me a parity complex should be defined so that the original statement of Theorem 4.2 holds for this definition. Also, Street's approach concerning his axioms seems to be "whatever works", so that morally tightness can be made part of the definition without too much concern.

@TimCampion: $\mu(x)$ is the graded set made of the $\mu(x)_p$.