Recent years have witnessed a growing academic interest in reconfigurable intelligent surface (RIS) since it is a promising technology to satisfy the performance requirements of six generation (6G). Works related to the RISs have been studied by numerous researchers, however, almost existing researches mainly focus on RIS-assisted MIMO systems with the assumption of uncorrelated environments. To date, far too little attention has been paid to situations that consider the correlations among the antennas and RIS reflecting elements. Therefore, in this paper, we focus on a RIS-assisted MIMO system under spatially correlated channels. We first analyze the ergodic spectral efficiency of the system and derive the upper bound of it. In particular, we propose an optimal phase shift design to maximize the ergodic spectral efficiency of the system. Numerical results show the effectiveness of our proposed phase shift design.