In this paper, we study the benefits of rate-splitting (RS) in multiple-input multiple-output (MIMO) cloud radio access networks (C-RAN). For this setting, we propose a stream-based transmission scheme in which user's messages are divided into a private and a common part, each of which is encoded into multiple streams. Under this stream-based strategy, we formulate a weighted sum-rate maximization problem subject to backhaul capacity and transmit power constraints. We determine the beamforming vectors of private and common streams of this non-convex optimization problem via a proposed iterative algorithm based on the fractional programming (FP) framework. Numerical results show gains of up to 27% of RS-MIMO over the baseline scheme of treating interference as noise (TIN). Particularly, at large backhaul capacities and specific antenna settings at which interference levels are maximal, rate splitting along with common message decoding is a viable option for effective interference management in MIMO C-RAN.