We consider a generalization of the rate-distortion-perception framework for lossy compression in which the reconstruction must attain a certain target distribution. This may arise, for example, in image restoration applications when there is a bit interface between a sender who records with degraded quality and receiver who wishes to recover a clear image. In this work, we characterize this as an optimal transport problem with constrained entropy between source and target distributions. We show that optimal solutions to this problem follow a framework that partially decouples the problems of compression and transport, and demonstrate the utility of common randomness. We show that the performance of a deep learning architecture following this framework is competitive with an end-to-end system.