We derive a three-dimensional shear-wave velocity model of the Ligurian-Provence back-arc basin (Northwestern Mediterranean Sea) using ocean-bottom seismometers (AlpArray OBSs) and land stations from permanent and temporary seismic networks. The quality of OBS continuous records is enhanced by a specific processing that reduces instrumental and seabed-induced noises (transients, tilt, compliance). To further improve the resolution of ambient-noise tomography in the offshore area, we compute the Rayleigh-wave part of the Green functions for OBS-OBS pairs by using onshore stations as virtual sources. 2-D group-velocity maps and their uncertainties are computed in the 4 - 150 s period range by a transdimensional inversion of Rayleigh-wave travel times. The dispersion data and their uncertainties are inverted for a probabilistic 3-D shear-wave velocity model that includes probability densities for Vs and for the depth of layer interfaces. The probabilistic model is refined by a linearized inversion that accounts for the water layer in the Ligurian Sea. Our S-wave velocity and layer boundary probability models correspond well to a recent, high-resolution P-wave velocity cross-section derived from controlled-source seismic profiling along the Ligurian-Provence basin axis. A joint interpretation of the P- and S-wave velocity sections along this profile reveals a thin, anomalous oceanic crust of low P-wave velocities but high S-wave velocities, intruded by a gabbroic body. The illuminated part of the upper mantle appears to be devoid of serpentinization.