Let $(X,d)$ be a compact metric space of finite diameter. Recall that we can always compute the Hausdorff distance between subsets $A, B \subset X$ via $$ d_H(A,B) = \max\left[\sup_{a\in A}\inf_{b\in B}d(a,b), \sup_{b\in B}\inf_{a\in A}d(a,b)\right]$$

Here is a simple question:

Is there a nice (pseudo-)metric structure on the set of all finite polyhedral decompositions of $X$?

For instance, if two such decompositions $P_1$ and $P_2$ had the same number of polyhedra, one could always use the infimum over all bijections $\phi: P_1 \to P_2$ of the largest Hausdorff distance between polyhedra $p \in P_1$ and $\phi(p) \in P_2$. But in general there is no reason to assume that $|P_1| = |P_2|$. Is there a standard way to deal with this general case?