A fast and rigorous hybrid time-domain simulator for analyzing wave interactions with complex structures involving electrically large platforms along with cable-interconnected antennas, shielding enclosures, and printed circuit boards is described. To efficiently simulate such complex structures, three different solvers are hybridized: A TDIE-based field solver for computing external fields on platforms, antennas, enclosures, boards, and cable shields; a MNA-based circuit solver for computing currents and voltages on circuits; and a TDIE-based cable solver for computing guided fields along cables. Field, circuit, and cable solvers are interfaced at cable terminations and along the cable shields, which results in a coupled system of equations that is solved simultaneously at each time step. Computation of the external and guided fields is accelerated using FFT-based algorithms and parallelization. The resulting solver permits the analysis of electrically large and geometrically intricate structures loaded with cables. The accuracy, efficiency, and versatility of the hybrid simulator are demonstrated by analyzing several EMC/EMI problems.