In the downlink of cloud radio access networks, a central encoder is connected to multiple multi-antenna base stations (BSs) via finite-capacity fronthaul links. At the central encoder, precoding is followed by compression in order to produce the rate-limited bit streams delivered to each BS. In current state-of-the-art schemes, the signals intended for different BSs are compressed independently. In contrast, this work proposes to leverage joint compression, also referred to as multivariate compression, of the signals for different BSs in order to better control the effect of the additive quantization noises at the mobile stations.