  Dirac fluids - interacting systems obeying particle-hole symmetry and Lorentz
invariance - are among the simplest hydrodynamic systems; they have also been
studied as effective descriptions of transport in strongly interacting Dirac
semimetals. Direct experimental signatures of the Dirac fluid are elusive, as
its charge transport is diffusive as in conventional metals. In this paper we
point out a striking consequence of fluctuating relativistic hydrodynamics: the
full counting statistics (FCS) of charge transport is highly non-gaussian. We
predict the exact asymptotic form of the FCS, which generalizes a result
previously derived for certain interacting integrable systems. A consequence is
that, starting from quasi-one dimensional nonequilibrium initial conditions,
charge noise in the hydrodynamic regime is parametrically enhanced relative to
that in conventional diffusive metals.
