Wide-swath altimetry, e.g. the Surface Water and Ocean Topography (SWOT) mission is expected to provide Sea Surface Height (SSH) measurements resolving scales of a few tens of kilometers. Over a large fraction of the globe, the SSH signal at these scales is essentially a superposition of a component due to balanced motions (BMs) and another component due to internal tides (ITs). Several oceanographic applications require the separation of these components and their mapping on regular grids. For that purpose, the paper introduces an alternating minimization algorithm that iteratively implements two data assimilation techniques, each specific to the mapping of one component: a quasi-geostrophic model with Back-and-Forth Nudging for BMs, and a linear shallow-water model with 4-Dimensional Variational (4DVar) assimilation for ITs. The algorithm is tested with Observation System Simulation Experiments (OSSE) where the truth is provided by a primitive-equation ocean model in an idealized configuration simulating a turbulent jet and mode-one ITs. The algorithm reconstructs almost 80% of the variance of BMs and ITs, the remaining 20% being mostly due to dynamics that cannot be described by the simple models used. Importantly, in addition to the reconstruction of stationary ITs, the amplitude and phase of nonstationary ITs are reconstructed. Sensitivity experiments show that the quality of reconstruction significantly depends upon the timing of observations. Although idealized, this study represents a step forward towards the disentanglement of BMs and ITs signals from real wide-swath altimetry data.