  Sterile neutrinos at the eV scale have long been studied in the context of
anomalies in short baseline neutrino experiments. Their cosmology can be made
compatible with our understanding of the early Universe provided the sterile
neutrino sector enjoys a nontrivial dynamics with exotic interactions, possibly
providing a link to the Dark Matter (DM) puzzle. Interactions between DM and
neutrinos have also been proposed to address the long-standing "missing
satellites" problem in the field of large scale structure formation. Motivated
by these considerations, in this paper we discuss realistic scenarios with
light steriles coupled to DM. We point out that within this framework active
neutrinos acquire an effective coupling to DM that manifests itself as a new
matter potential in the propagation within a medium of asymmetric DM. Assuming
that at least a small fraction of asymmetric DM has been captured by the Sun,
we show that a sizable region of the parameter space of these scenarios can be
probed by solar neutrino experiments, especially in the regime of small
couplings and light mediators where all other probes become inefficient. In the
latter regime these scenarios behave as familiar $3+1$ models in all channels
except for solar data, where a Solar Dark MSW effect takes place. Solar Dark
MSW is characterized by modifications of the most energetic $^8$B and CNO
neutrinos, whereas the other fluxes remain largely unaffected.
