At present, the mainstream solution for the control of wave energy converters (WECs), the model predictive control (MPC)-like method, faces a gap between simulation-based research and practical application. Two of the major difficulties are the online computation burden and the requirement of real-time wave information. In this study, a fast solving strategy is proposed at the level of quadratic programming (QP), where a tailored warm-start algorithm is designed and combined with an early stop technique. Simulations show that only one iteration of interior-point method (IPM) suffices to reach over 95% efficiency of the exact MPC, which significantly speeds up the computation. The fast strategy is then deployed on the real-time controller of a prototype WEC platform. During this implementation, the instantaneous wave force is estimated by a Kalman filter entirely based on the basic feedback signals: position, velocity, and the generator current, while future wave forces are predicted by an autoregressive model. The wave tank test confirms that the proposed fast MPC is capable of being executed at a high frequency, achieving stable operations within constraints, and reaching satisfactory energy efficiency under real wave-body interactions.