  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. Based on observations of $\alpha$
Ori, a new semi-empirical method to add a wind to hydrostatic model atmospheres
of RSGs was recently developed. We use this method of adding a 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 data of HD 95687 and V602 Car obtained with VLTI/AMBER. We
compute model intensities, spectra and visibilities for different mass-loss
rates using the radiative transfer code Turbospectrum. The models are convolved
to match the different spectral resolutions of the VLTI instruments, studying a
wavelength range of $1.8-5\,\mathrm{\mu m}$ corresponding to the $K$, $L$ and
$M$-bands for GRAVITY and MATISSE data. We compare the model spectra and
visibilities with the published VLTI/AMBER data. 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 to MARCS with simple radiative equilibrium dramatically improves
the agreement with the visibilities and the spectra. Our results reproduce
observed extended atmospheres up to several stellar radii. 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 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 data. We
provide temperature and density stratifications that succeed for the first time
in reproducing observed interferometric properties of RSG atmospheres.
