  We consider the random-field Heisenberg model, a paradigmatic model for
many-body localization (MBL), and add a Markovian dephasing bath coupled to the
Anderson orbitals of the model's non-interacting limit. We map this system to a
classical facilitated hopping model that is computationally tractable for large
system sizes, and investigate its dynamics. The classical model exhibits a
robust crossover between an ergodic (thermal) phase and a frozen (localized)
phase. The frozen phase is destabilized by thermal subregions (bubbles), which
thermalize surrounding sites by providing a fluctuating interaction energy and
so enable off-resonance particle transport. Investigating steady state
transport, we observe that the interplay between thermal and frozen bubbles
leads to a clear transition between diffusive and subdiffusive regimes. This
phenomenology both describes the MBL system coupled to a bath, and provides a
classical analogue for the many-body localization transition in the
corresponding quantum model, in that the classical model displays long local
memory times. It also highlights the importance of the details of the bath
coupling in studies of MBL systems coupled to thermal environments.
