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.]
[EDIT: I do have a strategy for trying to prove that this. I am thinking of induction on the dimension of $M$, using the handle structure associated with the second barycentric subdivision. But for the induction it would require proving a stronger statement in the case when $M$ is PL homeomorphic to a disk. Even if this works, I'd still be interested in a reference if anyone has one.]
