<p>Muddy continental shelves protect many vulnerable coastlines by dissipating storm-wave energy, yet their buffering capacity remains difficult to predict because the rheology of fluid mud is poorly represented in existing coastal models. Here we show that a physics-based model incorporating independently measured, shear-rate-dependent mud rheology can predict wave damping without calibration to wave attenuation observations. The model reproduces laboratory measurements of wave decay, velocity structure, and mud viscosity, and successfully predicts wave attenuation during a storm over the Atchafalaya inner shelf. Although an equivalent Newtonian viscosity reproduces similar bulk wave attenuation, it yields different internal pathways of energy dissipation. Shear-dependent rheology redistributes viscous dissipation across a wider range of shear conditions and modifies its vertical distribution within the mud layer. These results show that sediment rheology exerts a fundamental control on wave damping by muddy shelves and suggest that constant-viscosity approaches may misrepresent sediment transport and shoreline response during energetic wave events.</p><p></p>