  We construct an analytical theory of interplay between synchronizing effects
by common noise and by global coupling for a general class of smooth
limit-cycle oscillators. Both the cases of attractive and repulsive coupling
are considered. The derivation is performed within the framework of the phase
reduction, which fully accounts for the amplitude degrees of freedom. Firstly,
we consider the case of identical oscillators subject to intrinsic noise,
obtain the synchronization condition, and find that the distribution of phase
deviations always possesses lower-law heavy tails. Secondly, we consider the
case of nonidentical oscillators. For the average oscillator frequency as a
function of the natural frequency mismatch, limiting scaling laws are derived;
these laws exhibit the nontrivial phenomenon of frequency repulsion
accompanying synchronization under negative coupling. The analytical theory is
illustrated with examples of Van der Pol and Van der Pol--Duffing oscillators
and the neuron-like FitzHugh--Nagumo system; the results are also underpinned
by the direct numerical simulation for ensembles of these oscillators.
