Understanding how internal tides distort and lose coherence as they propagate through the ocean has been identified as a key issue for interpreting data from the upcoming wide-swath altimeter mission SWOT. This study addresses that issue through the analysis of numerical experiments where a low mode internal tide propagates through a quasi-geostrophic turbulent jet. We present a set of reduced equations based on vertical mode projections that describe the dynamics of the low-mode internal tides as they propagate through the slower turbulent field. Diagnostics of the terms responsible for the interaction between the wave and the slow circulation are computed from the numerical outputs. The large scale change of stratification, on top of eddies and jet meanders, contributes significantly to these interaction terms which is shown to be consistent with an independent scaling analysis. The sensitivity of interaction terms to a degradation of the slow field spatial and horizontal resolution indicates that current day observing systems (Argo network, altimetry) may lack spatial resolution in order to correctly predict internal tide evolution. The upcoming SWOT satellite mission may improve upon this situation. The number of vertical modes required to properly estimate interaction terms is discussed. These results advocate development of a simplified model based on solving a modest number of the reduced equations subject to a prescribed mesoscale field and internal tide sources. Such a model would constitute an inexpensive way to calculate the distortion and loss of coherence of low-mode internal tides as they propagate through mesoscale circulation.