<p>Spatial structure plays a crucial role in shaping evolutionary game dynamics in multi-strategy populations. However, the influence of spatial diffusion on players’ strategy evolution and interaction outcomes in non-zero-sum rock–paper–scissors (RPS) games with asymmetric payoffs has not been fully explored. This paper investigates an RPS evolutionary game in which the loser receives zero payoff. By incorporating spatial diffusion into replicator dynamics, we analyze the system under both homogeneous mixing and spatially structured environments. The main results can be summarized as follows: (i) in the homogeneous-mixing case, the system undergoes a Hopf bifurcation, leading from a stable equilibrium to persistent oscillations; (ii) spatial diffusion gives rise to Turing instability and Turing–Hopf bifurcation in certain parameter regimes, and numerical simulations reveal complex spatio-temporal patterns; (iii) in some parameter regions, chaotic dynamics are confirmed by positive maximum Lyapunov exponent, indicating increased uncertainty in strategic decision-making, and the system is in a disordered state; (iv) when the winner’s payoff exceeds twice the tie payoff, the homogeneous equilibrium corresponding to an equal distribution of the three strategies is stable in both the well-mixed system and the reaction-diffusion system. These results indicate that spatial diffusion can fundamentally alter evolutionary outcomes and induce complex and chaotic strategy dynamics. The proposed framework provides a reference for the analysis of multi-strategy evolutionary games with spatial diffusion.</p>