The migrating diurnal tide is one of the dominant dynamical features in the low latitudes of the Earth's mesosphere and lower thermosphere (MLT) region, representing the atmospheric response to the largest component of solar forcing. Ground-based observations of the tide have resolved short-term variations attributed to nonlinear interactions between the tide and planetary waves that are also in the region. Using the NCAR Thermosphere Ionosphere Mesosphere Electrodynamics General Circulation Model (TIME-GCM), we simulate a quasi 2 day wave (QTDW) event under late-January conditions. In this case, sideband sum and difference child waves are resolved, indicating that a nonlinear interaction is occurring between the QTDW and the tide. The migrating diurnal tide in the MLT displays local amplitude decreases of 20-40%, as well as a shortening of vertical wavelength by roughly 4 km. Examining the physical mechanisms driving the interaction, nonlinear advection is found to result in amplification of the tide in some regions and damping in others, manifesting as increased smoothing of the tidal structure when the QTDW is present in the MLT. Additionally, the QTDW also enhances the easterly summer mean wind jet that can also account for changes in tidal amplitude and vertical wavelength. We find that QTDW-induced background atmosphere changes in TIME-GCM can drive tidal variability at levels greater than nonlinear advection, a possibility not previously considered.