Recall that algebras (or linear 1-categories) $A$ and $B$ are Morita equivalent if there exist bimodules $_AM_B$ and $_BN_A$ and isomorphisms $u: {}_A(M \otimes_BN)_A \to {}_AA_A$ and $v: {}_B(N \otimes_AM)_B \to {}_BB_B$ such that $u$ and $v$ satisfy the zig-zag relations.

I'm interested in what can said under the weaker hypothesis that ${}_B(N \otimes_AM)_B \cong {}_BB_B$ but ${}_A(M \otimes_BN)_A$ is not necessarily isomorphic to ${}_AA_A$.

One easy consequence is that the isomorphism classes of $B$-modules inject into the isomorphism classes of $A$-modules. Can the $A$-modules which do not come from $B$-modules be understood in terms of the bimodule ${}_A(M \otimes_BN)_A$?

In the situation I'm mainly interested in, $A$ and $B$ are *-algebras (actually *-categories) and $_AM_B$ is the same as $_BN_A$ under the usual identification of left and right modules.