This paper proposes an adaptive control law for the causal approximation of rate-independent linear damping (RILD). RILD provides direct control over displacement, a desirable feature for low-frequency structures. When low-frequency structures are subjected to high-frequency ground motions, RILD mitigates large displacements while avoiding large damping forces and floor accelerations. However, in ideal RILD, the force is proportional to the displacement advanced in phase π/2 radians, a non-causality that has limited its practical applications. An adaptive controller is proposed to approximate ideal RILD based on the frequency of vibration estimated in real-time and a filter-based causal model for RILD. By estimating the frequency of vibration, the phase advance is more accurately applied. The desired damping force is then tracked by a semi-active damper, which is naturally in phase with velocity and has a controllable magnitude. The adaptive control approach is demonstrated through the real-time hybrid simulation (RTHS) of a 14-story inter-story isolated building. An MR damper is added to the isolation layer of the structure to provide supplemental control with the goal of mimicking ideal RILD. The MR damper is experimentally represented while the remainder of the structure is numerically simulated in the RTHS loop. The results compare well to non-causal numerical simulations in both damping forces and structural responses. Results show improved seismic performance for the adaptive algorithm as compared to non-adaptive causal approximations of RILD.