Bases for pseudovarieties closed under bideterministic product

We show that if $\mathsf V$ is a semigroup pseudovariety containing the finite semilattices and contained in $\mathsf {DS}$, then it has a basis of pseudoidentities between finite products of regular pseudowords if, and only if, the corresponding variety of languages is closed under bideterministic product. The key to this equivalence is a weak generalization of the existence and uniqueness of $\mathsf J$-reduced factorizations. This equational approach is used to address the locality of some pseudovarieties. In particular, it is shown that $\mathsf {DH}\cap\mathsf {ECom}$ is local, for any group pseudovariety $\mathsf H$.