A novel hydrostatic coupled-mode shallow water model (CSW) is developed and used to simulate tides in the greater Middle Atlantic Bight region. The model incorporates realistic stratification and topography, an internal tide generating function (ITGF) that provides internal tide forcing from existing surface tide parameters, and dynamical terms that describe linearized wave- mean-flow and mean-density interactions. Several idealized and realistic simulations are used to verify the model. These verification simulations include internal-tide interactions involving topographic coupling and mean-flow coupling, and comparisons with other simpler and more complex nonlinear primitive-equation models. Then, twenty-four simulations of internal tide generation and propagation in the greater Middle Atlantic Bight region are used to identify significant internal-tide interactions with the Gulf Stream. The simulations indicate that locally generated mode-1 internal tides can refract and/or reflect at the Gulf Stream. The redirected internal tides often re-appear at the shelfbreak, where they produce onshore energy fluxes that are intermittent (i.e., noncoherent) because meanders in the Gulf Stream alter their precise location, phase, and amplitude. These results provide an explanation for the anomalous onshore energy fluxes previously observed at the New Jersey Shelfbreak and linked with the generation of nonlinear internal waves.