[EDIT: I now think I can see how to prove that this can be done by induction on the dimension of $M$, using the handle structure associated with the second barycentric subdivision. But I'd still be interested in a reference if anyone has one.]

If $P$ and $Q$ are compact codimension zero submanifolds of a PL manifold, say that they meet nicely if $P\cap Q$ is a codimension zero submanifold of both $\partial P$ and $\partial Q$. In particular then $P\cup Q$ is a codimension zero submanifold.

Question: Given a triangulation of a compact PL manifold $M$, do there always exist compact codimension zero PL submanifolds $M_1,\dots ,M_k$ of $M$ such that:

(1) $M=M_1\cup\dots\cup M_k$,

(2) for every $i$, $M_1\cup\dots\cup M_{i-1}$ meets $M_i$ nicely, and

(3) for every $i$, $M_i$ is contained in some simplex of the given triangulation of $M$.

[Note: These submanifolds are not going to be subcomplexes for the given triangulation. They will be subcomplexes for some finer trangulation. But it is required that each $M_i$ is a subset of some original simplex.]