  We present an effective field theory for the nonlinear fluctuating
hydrodynamics of a single conserved charge with or without time-reversal
symmetry, based on the Martin-Siggia-Rose formalism. Applying this formalism to
fluids with only charge and multipole conservation, and with broken
time-reversal symmetry, we predict infinitely many new dynamical universality
classes, including some with arbitrarily large upper critical dimensions. Using
large scale simulations of classical Markov chains, we find numerical evidence
for a breakdown of hydrodynamics in quadrupole-conserving models with broken
time-reversal symmetry in one spatial dimension.
