  We study the collective dynamics of identical phase oscillators on globally
coupled networks whose interactions are asymmetric and mediated by positive and
negative couplings. We split the set of oscillators into two interconnected
subpopulations. In this setup, oscillators belonging to the same group interact
via symmetric couplings while the interaction between subpopulations occurs in
an asymmetric fashion. By employing the dimensional reduction scheme of the
Ott-Antonsen (OA) theory, we verify the existence of traveling-wave and
$\pi$-states, in addition to the classical fully synchronized and incoherent
states. Bistability between all collective states is reported. Analytical
results are generally in excellent agreement with simulations; for some
parameters and initial conditions, however, we numerically detect chimera-like
states which are not captured by the OA theory.
