Multi-domain translation seeks to learn a probabilistic coupling between marginal distributions that reflects the correspondence between different domains. We assume that data from different domains are generated from a shared latent representation based on a structural equation model. Under this assumption, we show that the problem of computing a probabilistic coupling between marginals is equivalent to learning multiple uncoupled autoencoders that embed to a given shared latent distribution. In addition, we propose a new framework and algorithm for multi-domain translation based on learning the shared latent distribution and training autoencoders under distributional constraints. A key practical advantage of our framework is that new autoencoders (i.e., new domains) can be added sequentially to the model without retraining on the other domains, which we demonstrate experimentally on image as well as genomics datasets.