Weak Frobenius monads and Frobenius bimodules

As shown by S. Eilenberg and J.C. Moore (1965), for a monad $F$ with right adjoint comonad $G$ on any catgeory $\mathbb{A}$, the category of unital $F$-modules ${\mathbb{A}}_F$ is isomorphic to the category of counital $G$-comodules ${\mathbb{A}}^G$. The monad $F$ is Frobenius provided we have $F=G$ and then ${\mathbb{A}}_F\simeq {\mathbb{A}}^F$. Here we investigate which kind of equivalences can be obtained for non-unital monads (and non-counital comonads).