The non persistent phase relationship between internal tides and astronomical forcings, also known as incoherence, has been identified as a major question in the context of future wide-swath satellite altimetry. This study addresses this issue using a novel set of numerical experiments where a plane-wave/low-mode internal tide propagates through a turbulent mesoscale eddy field. These experiments demonstrate the emergence of internal tide incoherence as the eddy turbulence is strengthened. In strongly turbulent situations, the internal tide signature on sea level forms complex interference patterns with large amplifications of the initial internal wave. These patterns evolve more rapidly than the signature of the turbulent eddy field on sea level. The implications of such idealized numerical simulations for wide-swath altimetry are discussed.