  We study the solid-to-liquid transition in a two-dimensional fully periodic
soft-glassy model with an imposed spatially heterogeneous stress. The model we
consider consists of droplets of a dispersed phase jammed together in a
continuous phase. When the peak value of the stress gets close to the yield
stress of the material, we find that the whole system intermittently tunnels to
a metastable "fluidized" state, which relaxes back to a metastable "solid"
state by means of an elastic-wave dissipation. This macroscopic scenario is
studied through the microscopic displacement field of the droplets, whose time
statistics displays a remarkable bimodality. Metastability is rooted in the
existence, in a given stress range, of two distinct stable rheological branches
as well as long-range correlations (e.g., large dynamic heterogeneity)
developed in the system. Finally, we show that a similar behavior holds for a
pressure-driven flow, thus suggesting possible experimental tests.
