  We show that disordered boundaries destroy bulk phase separation in scalar
active systems in dimension $d<d_c=3$. This is in strong contrast with the
equilibrium case where boundaries have no impact on the bulk of phase-separated
systems. The underlying mechanism is revealed by considering a localized
deformation of an otherwise flat wall, from which the case of a disordered
boundary can be inferred. We find long-ranged correlations of the density field
as well as a cascade of eddies which we show prevent bulk phase separation in
low enough dimensions. The results are derived for dilute systems as well as in
the presence of interactions, under the sole condition that the density field
is the unique hydrodynamic mode. Our theoretical calculations are validated by
numerical simulations of microscopic active systems.
