Recent studies have explored the synergy between reconfigurable intelligent surfaces (RIS) and multiple-input multiple-output (MIMO) systems to overcome limitations in spectral efficiency and coverage. While RIS-aided MIMO systems promise transformative gains, their real-world deployment faces critical challenges rooted in hardware imperfections, in particular, the nonlinearity of power amplifiers (PAs) in the MIMO transmitters. To bridge this gap, this paper investigates a holistic design that jointly optimizes RIS phase shifts and base station transmit precoder for the RIS-aided MIMO systems with a practical PA model in real life. We propose a hardware-aware joint design framework that co-optimizes the base station precoder and RIS phase shifts to balance beamforming gain with PA distortion suppression. A closed-form solution for the optimal RIS phase-shifting matrix is derived as a function of the precoder, reducing the joint optimization to an equivalent precoder optimization problem. A low-complexity successive convex approximation-based algorithm is developed for efficient precoder design. Simulations demonstrate that the proposed method achieves up to 30.6% spectral efficiency improvement over benchmarks under PA nonlinearity, highlighting its practical significance in RIS-aided systems.