  We consider a population of globally coupled oscillators in which phase
shifts in the coupling are random. We show that in the maximally disordered
case, where the pairwise shifts are i.i.d. random variables, the dynamics of a
large population reduces to one without randomness in the shifts but with an
effective coupling function, which is a convolution of the original coupling
function with the distribution of the phase shifts. This result is valid for
noisy oscillators and/or in presence of a distribution of natural frequencies.
We argue also, using the property of global asymptotic stability, that this
reduction is valid in a partially disordered case, where random phase shifts
are attributed to the forced units only. However, the reduction to an effective
coupling in the partially disordered noise-free situation may fail if the
coupling function is complex enough to ensure the multistability of locked
states.
