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We propose a novel distributional regression model for a multivariate response vector based on a copula process over the covariate space. It uses the implicit copula of a Gaussian multivariate regression, which we call a ``regression copula''. To allow for large covariate vectors their coefficients are regularized using a novel multivariate extension of the horseshoe prior. Bayesian inference and distributional predictions are evaluated using efficient variational inference methods, allowing application to large datasets. An advantage of the approach is that the marginal distributions of the response vector can be estimated separately and accurately, resulting in predictive distributions that are marginally-calibrated. Two substantive applications of the methodology highlight its efficacy in multivariate modeling. The first is the econometric modeling and prediction of half-hourly regional Australian electricity prices. Here, our approach produces more accurate distributional forecasts than leading benchmark methods. The second is the evaluation of multivariate posteriors in likelihood-free inference (LFI) of a model for tree species abundance data, extending a previous univariate regression copula LFI method. In both applications, we demonstrate that our new approach exhibits a desirable marginal calibration property.