The Lagrangian and Eulerian near-surface current signatures of a low-mode internal tide propagating through a turbulent jet are compared in an idealized numerical simulation. We estimate and compare internal tides’ stationary and nonstationary velocity amplitudes as well as non-stationarity timescales. We find Lagrangian internal tides total amplitude similar to Eulerian one. Lagrangian velocity are mostly nonstationary and Lagrangian non-stationary timescales are comparable to or smaller than Eulerian ones. This low-bias is proposed to be the result of the deformation of internal tide surface signal along the drift induced by lower frequency surface currents. A model based on the latter hypothesis successfully predicts Lagrangian autocovariance and highlights its dependence to Eulerian autocovariance and to the properties of the internal tides and jet. We address the implications of these results in the context of the Surface Water and Ocean Topography (SWOT) mission, for which the separation of mesoscale balanced flow and internal tides with data at the ocean’s surface was raised.