  We study the effect of common noise on coupled active rotators. While such a
noise always facilitates synchrony, coupling may be attractive or repulsing. We
develop an analytical approach based on a transformation to approximate
angle-action variables and averaging over fast rotations. For identical
rotators, we describe a transition from full to partial synchrony at a critical
value of repulsive coupling. For nonidentical rotators, the most nontrivial
effect occurs at moderate repulsive coupling, where a juxtaposition of phase
locking with frequency repulsion (anti-entrainment) is observed. We show that
the frequency repulsion obeys a nontrivial power law.
