  We study non-equilibrium analogues of surface phase transitions in a minimal
model of active particles in contact with a purely repulsive potential barrier
that mimics a thin porous membrane. Under conditions of bulk motility-induced
phase separation, the interaction strength $\varepsilon_w$ of the barrier
controls the affinity of the dense phase for the barrier region. We uncover
clear signatures of a wetting phase transition as $\varepsilon_w$ is varied. In
common with its equilibrium counterpart, the character of this transition
depends on the system dimensionality: a continuous transition with large
density fluctuations and gas bubbles is uncovered in 2d while 3d systems
exhibit a sharp transition absent of large correlations.
