  The simplest network of coupled phase-oscillators exhibiting chimera states
is given by two populations with disparate intra- and inter-population coupling
strengths. We explore the effects of heterogeneous coupling phase-lags between
the two populations. Such heterogeneity arises naturally in various settings,
for example as an approximation to transmission delays, excitatory-inhibitory
interactions, or as amplitude and phase responses of oscillators with
electrical or mechanical coupling. We find that breaking the phase-lag symmetry
results in a variety of states with uniform and non-uniform synchronization,
including in-phase and anti-phase synchrony, full incoherence (splay state),
chimeras with phase separation of $0$ or $\pi$ between populations, and states
where both populations remain desynchronized. These desynchronized states
exhibit stable, oscillatory, and even chaotic dynamics. Moreover, we identify
the bifurcations through which chimeras emerge. Stable chimera states and
desynchronized solutions, which do not arise for homogeneous phase-lag
parameters, emerge as a result of competition between synchronized in-phase,
anti-phase equilibria, and fully incoherent states when the phase-lags are near
$\pm\frac{\pi}{2}$ (cosine coupling). These findings elucidate previous
experimental results involving a network of mechanical oscillators and provide
further insight into the breakdown of synchrony in biological systems.
