  We investigate the phenomenon of second sound in various states of matter
from the perspective of non-equilibrium effective field theory (EFT). In
particular, for each state of matter considered, we find that at least two
(though sometimes multiple) qualitatively different EFTs exist at finite
temperature such that there is always at least one EFT with a propagating
second-sound wave and at least one with no such second-sound wave. To aid in
the construction of these EFTs, we use the method of cosets developed for
non-equilibrium systems. It turns out that the difference between the EFTs with
and without second-sound modes can be understood as arising from different
choices of a new kind of inverse Higgs constraint. Finally, we demonstrate that
it is possible to bypass the need for new inverse Higgs constraints by
formulating EFTs on a new kind of manifold that is like the usual fluid
worldvolume, but with reduced gauge symmetries.
