This is equivalent to my earlier question https://mathoverflow.net/questions/159473/a-question-about-something-like-shelling-in-a-pl-manifold, but maybe more comprehensible and to the point.

Given a triangulation of the PL sphere $S^n$, is there always a subdivision (a.k.a. refinement, a.k.a. finer triangulation) that makes it shellable?

Put this way, I'm guessing that the answer is well-known.

EDIT: I quickly got two different answers, each of which seems to give just what I need. I'm more or less arbitrarily accepting Allan's.