In this contribution, a practical reconfigurable intelligent surface (RIS)-assisted multiple-input multiple-output (MIMO) system is presented to improve the performance and spectral efficiency (SE) of state-of-the-art RIS-assisted spatial modulation systems by introducing phase offsets on the RIS and transmitter. As opposed to conventional RIS-assisted spatial modulation schemes, the proposed RIS-assisted generalized spatial modulation with phase offset (RIS-GSM-P) scheme activates an arbitrary number of transmit antennas to form a wider variety of transmit antenna combinations (TAC) for improving SE. Depending on whether the direct path is available or not, two phase offset schemes are conceived to enhance the beamforming of the RIS and to widen the channel difference between TACs. Finally, a unified framework for the theoretical bit error rate upper bound of the proposed RIS-GSM-P scheme is presented, which is also compatible with several conventional RIS-assisted transmission schemes. Furthermore, a low-complexity detector is provided to reduce the detection complexity. Monte Carlo simulation results are provided to demonstrate the effectiveness of the proposed schemes and to validate our theoretical analysis.