In the downlink of cloud radio access networks, a central encoder is connected to multiple multi-antenna base stations (BSs) via finite-capacity backhaul links. At the central encoder, precoding is followed by compression in order to produce the rate-limited bit streams delivered to each BS over the corresponding backhaul link. 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 (MSs). The problem of maximizing the weighted sum-rate over precoding and compression strategies is formulated subject to power and backhaul capacity constraints. An iterative algorithm is proposed that achieves a stationary point of the problem. From numerical results, it is confirmed that the proposed joint precoding and compression strategy outperforms conventional approaches based on independent compression across the BSs.