Multidomain and multilingual machine translation often rely on parameter sharing strategies, where large portions of the network are meant to capture the commonalities of the tasks at hand, while smaller parts are reserved to model the peculiarities of a language or a domain. In adapter-based approaches, these strategies are hardcoded in the network architecture, independent of the similarities between tasks. In this work, we propose a new method to better take advantage of these similarities, using a latent-variable model. We also develop new techniques to train this model end-to-end and report experimental results showing that the learned patterns are both meaningful and yield improved translation performance without any increase of the model size.