The Lagrangian and Eulerian surface current signatures of a low-mode internal tide propagating through a turbulent balanced flow are compared in idealized numerical simulations. Lagrangian and Eulerian total (i.e., coherent plus incoherent) tidal amplitudes are found to be similar. Compared to Eulerian diagnostics, the Lagrangian tidal signal is more incoherent with comparable or smaller incoherence time scales and larger incoherent amplitudes. The larger level of incoherence in Lagrangian data is proposed to result from the deformation of an Eulerian internal tide signal induced by drifter displacements. Based on the latter hypothesis, a theoretical model successfully predicts Lagrangian autocovariances by relating Lagrangian and Eulerian autocovariances and the properties of the internal tides and jet. These results have implications for the separation of balanced flow and internal tides signals in the sea level data collected by the future Surface Water and Ocean Topography (SWOT) satellite mission.