coarser than triangulations “almost partitions” into simplices

The total space $T$ of an embedded into $\mathbb{R}^n$ pure $n$-dimensional simplicial complex (in other words, the union of finitely many $n$-dimensional compact convex polytopes) sometimes admits an "almost partition" into $n$-simplices (i.e. $T$ equals the union of these simplices, and the interiors of these simplices do not, pairwise, intersect) which is coarser than a triangulation.

For instance, consider the union $T$ of two tetrahedra $T_1$, $T_2$ in $\mathbb{R}^3$ which intersect at the mid-point $M=E_1\cap E_2$ of edges $E_i\in T_i$, $1\leq i\leq 2$. Then any triangulation of $T$ will have to include $M$, and have at least 4 simplices. On the other hand, there exist an obvious "almost partition" of $T$ into just 2 simplices, $T_1$ and $T_2$.

We wonder whether there exists any research on these kinds of "almost partitions", and whether there is any well-established terminology here.

-