For $H$ a Hopf-algebra, an $H$-Hopf-Galois extension is a map of rings $\phi\colon\thinspace A\to B$ such that $H$ coacts on $B$ over $A$, $B\otimes_AB\cong B\otimes H$, and the cofixed points, or the primitives, of the $H$-coaction on $B$ are isomorphic to $A$, i.e. $A\cong B^{coH}$, where $B^{coH}$ is the equalizer of the coaction and the unit map of $H$ tensored up to $B$.

However, I seem to be running into some situations where these properties all hold, but $B$ is not a ring. Rather, it's only a unital $A$-module, with the unit map being exactly that map I described. None of those requirements necessarily require $B$ to be a ring (I guess we'd better be careful about what category those isomorphisms live in though, so maybe you can only recover $A$ as a unital $A$-module or something).

One example of this is when you've got an action of an $A_\infty$-space $G$ on an $A_\infty$-algebra $R$. Then the quotient $R/G$ is not necessarily still $A_\infty$, but by Koszul duality, it has a $BG$-coaction that you can use to recover some information about $R$.

My only real question, since (I think) I'm running into this in the wild, is, does this kind of structure have a name? Additionally, does this kind of structure have any geometric interpretation? When $B$ is a ring, we can think of it as saying that we've got a principal $Spec(H)$-bundle $Spec(B)\to Spec(A)$ (when everything is noncommutative these are even sometimes called quantum principal bundles). But there doesn't seem to be any immediate geometric significance if $B$ is not a ring.