  Phase synchronization between collective oscillations exhibited by two weakly
interacting groups of non-identical phase oscillators with internal and
external global sinusoidal coupling of the groups is analyzed theoretically.
Coupled amplitude equations describing the collective oscillations of the
oscillator groups are obtained by using the Ott-Antonsen ansatz, and then
coupled phase equations for the collective oscillations are derived by phase
reduction of the amplitude equations. The collective phase coupling function,
which determines the dynamics of macroscopic phase differences between the
groups, is calculated analytically. It is demonstrated that the groups can
exhibit effective anti-phase collective synchronization even if the microscopic
external coupling between individual oscillator pairs belonging to different
groups is in-phase, and similarly effective in-phase collective synchronization
in spite of microscopic anti-phase external coupling between the groups.
