Extending Doi--Takeuchi correspondence to bimodules and noncommutative covers

Doi and Takeuchi, and independently Blattner, Cohen and Montgomery, have exhibited a correspondence between cleft extensions of an algebra $U$ by a Hopf algebra $H$ and $H$-measurings on $U$ equipped with a compatible normalized 2-cocycle. Each cleft extension is isomorphic as an algebra, a left $U$