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Electromagnetic (EM) wave scattering in electrically large, irregularly shaped, environments is a common phenomenon. The deterministic, or first principles, study of this process is usually computationally expensive and the results exhibit extreme sensitivity to scattering details. For this reason, the deterministic approach is often dropped in favor of a statistical one. The Random Coupling Model (RCM) is one such approach that has found great success in providing a statistical characterization for wave chaotic systems in the frequency domain. Here we aim to transform the RCM into the time domain and generalize it to new situations. The proposed time-domain RCM (TD-RCM) method can treat a wave chaotic system with multiple ports and modes. Two features are now possible with the time-domain approach for chaotic resonators: the incorporation of earlytime short-orbit transmission path effects between the ports, and the inclusion of arbitrary nonlinear or time-varying port load impedances. We have conducted short-pulse time-domain experiments in wave chaotic enclosures, and used the TD-RCM to simulate the corresponding experimental setup. We have also examined a diode-loaded port and compared experimental results with a numerical TD-RCM treatment and found agreement.