  We develop a theory of Coulomb drag in ultraclean double layers with strongly
correlated carriers. In the regime where the equilibration length of the
electron liquid is shorter than the interlayer spacing the main contribution to
the Coulomb drag arises from hydrodynamic density fluctuations. The latter
consist of plasmons driven by fluctuating longitudinal stresses, and diffusive
modes caused by temperature fluctuations and thermal expansion of the electron
liquid. We express the drag resistivity in terms of the kinetic coefficients of
the electron fluid. Our results are nonperturbative in interaction strength and
do not assume Fermi-liquid behavior of the electron liquid.
