Poset block equivalence of integral matrices

Given square matrices $B$ and $B'$ with a poset-indexed block structure (for which an $ij$ block is zero unless $i\preceq j$), when are there invertible matrices $U$ and $V$ with this required-zero-block structure such that $UBV = B'$? We give complete invariants for the existence of such an equivalence for matrices over a principal ideal domain $\mathcal R$. As one application, when $\mathcal R$ is a field we classify such matrices up to similarity by matrices respecting the block structure. We also give complete invariants for equivalence under the additional requirement that the diagonal blocks of $U$ and $V$ have determinant $1$. The invariants involve an associated diagram (the ``$K$-web'') of $\mathcal R$-module homomorphisms. The study is motivated by applications to symbolic dynamics and $C^*$-algebras.