  Relaxation of quantum systems is a central problem in nonequilibrium physics.
In contrast to classical systems, the underlying quantum dynamics results not
only from atomic interactions but also from the long-range coherence of the
many-body wave function. Experimentally, nonequilibrium states of quantum
fluids are usually created using moving objects or laser potentials, directly
perturbing and detecting the system's density. However the fate of long-range
phase coherence for hydrodynamic motion of disordered quantum systems is less
explored, especially in three dimension. Here, we unravel how the density and
phase coherence of a Bose-Einstein condensate of $^6$Li$_2$ molecules respond
upon quenching on or off an optical speckle potential. We find that, as the
disorder is switched on, long-range phase coherence breaks down one order of
magnitude faster than the density of the quantum gas responds. After removing
it, the system needs two orders of magnitude longer times to reestablish
quantum coherence, compared to the density response. We compare our results
with numerical simulations of the Gross-Pitaevskii equation on large
three-dimensional grids, finding an overall good agreement. Our results shed
light on the importance of long-range coherence and possibly long-lived phase
excitations for the relaxation of nonequilibrium quantum many-body systems.
