  We study the coherent collective diffusive motion of neraly phase-ordered
active oscillators on a substrate. We construct the hydrodynamic theory of this
system, and use it to make many testable predictions. We find that for a wide
range of parameters this system can be stable with orientational order
generically stronger or weaker than quasi long range order (QLRO) in different
regions of the parameter space. The number density fluctuations correspondingly
are either generally hyperuniform, or giant. In the remaining parameter space,
the system becomes unstable. Simulations of an equivalent agent-based
lattice-gas model support the predictions of the hydrodynamic theory on the
phase-ordered states. Our theory forms the basis for theoretical understanding
of wide-ranging phenomena, e.g., oscillator synchronization in networks with
evolving connections and active superfluids.
