Context. Red supergiants (RSGs) are evolved massive stars in a stage preceding core-collapse supernova. The physical processes that trigger mass loss in their atmospheres are still not fully understood, and they remain one of the key questions in stellar astrophysics. Based on observations of α Ori, a new semi-empirical method to add a wind to hydrostatic model atmospheres of RSGs was recently developed. This method can reproduce many of the static molecular shell (or ‘MOLsphere’) spectral features.Aims. We used this method of adding a semi-empirical wind to a MARCS model atmosphere to compute synthetic observables, comparing the model to spatially resolved interferometric observations. We present a case study to model published interferometric data of HD 95687 and V602 Car obtained with the AMBER instrument at the Very Large Telescope Interferometer (VLTI).Methods. We computed model intensities with respect to the line-of-sight angle (µ) for different mass-loss rates, spectra, and visibilities using the radiative transfer code TURBOSPECTRUM. We were able to convolve the models to match the different spectral resolutions of the VLTI instruments, studying a wavelength range of 1.8–5 µm corresponding to the K, L, and M bands for GRAVITY and MATISSE data. The model spectra and squared visibility amplitudes were compared with the published VLTI/AMBER data.Results. The synthetic visibilities reproduce observed drops in the CO, SiO, and water layers that are not shown in visibilities based on MARCS models alone. For the case studies, we find that adding a wind onto the MARCS model with simple radiative equilibrium dramatically improves the agreement with the squared visibility amplitudes as well as the spectra, with the fit being even better when applying a steeper density profile than predicted from previous studies. Our results reproduce observed extended atmospheres up to several stellar radii.Conclusions. This paper shows the potential of our model to describe extended atmospheres in RSGs. It can reproduce the shapes of the spectra and visibilities with a better accuracy in the CO and water lines than previous models. The method can be extended to other wavelength bands for both spectroscopic and interferometric observations. We provide temperature and density stratifications that succeed, for the first time, in reproducing observed interferometric properties of RSG atmospheres.Key words: stars: atmospheres / stars: massive / stars: evolution / stars: fundamental parameters / supergiants / stars: mass-loss★ Full spectral resolution models are only available at the CDS via anonymous ftp to cdsarc.cds.unistra.fr (130.79.128.5) or via https://cdsarc.cds.unistra.fr/viz-bin/cat/J/A+A/669/A76