  Recent experimental and theoretical efforts have focused on the effect of
dissipation on quantum many-body systems in their many-body localized (MBL)
phase. While in the presence of dephasing noise such systems reach a unique
ergodic state, their dynamics is characterized by slow relaxation manifested in
non-exponential decay of self-correlations. Here we shed light on a currently
much debated issue, namely the role of interactions for this relaxation
dynamics. We focus on the experimentally relevant situation of the evolution
from an initial charge density wave in the presence of strong dephasing noise.
We find a crossover from a regime dominated by disorder to a regime dominated
by interactions, with a concomitant change of time correlators from stretched
exponential to compressed exponential form. The strongly interacting regime can
be explained in terms of nucleation and growth dynamics of relaxing regions -
reminiscent of the kinetics of crystallization in soft matter systems - and
should be observable experimentally. This interaction-driven crossover suggests
that the competition between interactions and noise give rise to a much richer
structure of the MBL phase than anticipated so far.
