Ensemble weather forecasts typically exhibit systematic errors and require post-processing to obtain reliable forecasts. Accurately modeling multivariate dependencies is crucial in many practical applications, and various multivariate post-processing approaches have been proposed, where ensemble predictions are first post-processed separately in each margin, and multivariate dependencies are then restored via copulas. These methods share common limitations, especially the difficulty to incorporate additional predictors. We propose a novel method based on generative machine learning to address these challenges. In this new class of nonparametric distributional regression models, samples from the multivariate forecast distribution are directly obtained as output of a generative model.