  A mesoscopic theory for water-in-salt electrolytes combining density
functional and field-theoretic methods is developed in order to explain the
unexpectedly large period of the oscillatory decay of the disjoining pressure
observed in recent experiments for the LiTFSI (lithium
bis(trifluoromethylsulfonyl)-imide) salt [T. S. Groves et. al., J. Phys. Chem.
Lett. {\bf 12},1702 (2021)]. We assumed spherical ions with different
diameters, and implicit solvent inducing strong, short-range attraction between
ions of the same sign. For this highly simplified model, we calculated
correlation functions. Our results indicate that mesoscopic inhomogeneities can
occur when the sum of the Coulomb and the water-mediated interactions between
like ions is attractive at short- and repulsive at large distances. We adjusted
the attractive part of the potential to the water-in-LiTFSI electrolyte, and
obtained both the period and the decay rate of the correlations in a
semiquantitative agreement with the experiment. In particular, the decay length
of the correlations increases nearly linearly with the volume fraction of ions.
