Statistical post-processing of ensemble forecasts usually is carried out independently for individual, scalar predictands. However, in some applications multivariate joint forecast distributions, which capture both the univariate marginal distributions of their constituent scalar predictands as well as the dependence structure among them, may be required. Copulas are functions that link multivariate distribution functions to their constituent univariate marginal distributions. Empirical copulas are non-parametric copula functions that are easy to implement. This article compares recently proposed variants of empirical copula coupling (ECC-Q and ECC-R), which take their dependence structures from raw forecast ensembles, and the Schaake shuffle, which is based on unconditional random samples from the historical climatology, in a four-dimensional multivariate ensemble post-processing setting. These alternatives were compared for probability forecasts of multi-day ‘heat waves’, based on the 11-member National Oceanic and Atmospheric Administration (NOAA) reforecast ensembles. Best forecast accuracy was achieved using the unconditional climatological dependence structures sampled by the Schaake shuffle, implying that any forecast improvements due to flow-specific dependencies that might be captured by the ensemble-based copulas are not sufficient to overcome errors in the ensemble's representation of those dependencies.