Equivalent bimodule categories

Question

Let $A,B$ be two rings such that their categories of bimodules are equivalent: $$A\mathsf{-Bimod} \simeq B\mathsf{-Bimod}$$ What can we say about $A$ and $B$? Are they isomorphic? Are they Morita-equivalent? Probably this has been studied in the literature, so this is primarily a reference request.

Answer 1

Let me work over a field $k$ and talk about bimodules over $k$ and $k$-linear categories (so we ask for an equivalence respecting the $k$-linear structure). Then it does not follow that $A$ and $B$ are Morita equivalent (or isomorphic) over $k$.

The given condition is equivalent to the condition that $A \otimes_k A^{op}$ and $B \otimes_k B^{op}$ are Morita equivalent. If $A, B$ are both central simple algebras over $k$ then both of these algebras are Morita equivalent to $k$, but $A$ and $B$ themselves are Morita equivalent iff they represent the same class in the Brauer group $\text{Br}(k)$. So, for example, we can take the two representatives $\mathbb{R}, \mathbb{H}$ of the Brauer group $\text{Br}(\mathbb{R})$.

On the other hand, if $A$ and $B$ are Morita equivalent, then $\text{Bimod}(A)$ and $\text{Bimod}(B)$ are equivalent even monoidally, because they can be described as the monoidal categories of $k$-linear cocontinuous endofunctors of $\text{Mod}(A)$ and $\text{Mod}(B)$ respectively by Eilenberg-Watts.