  We present the hydrodynamic theory of active XY spins coupled with flow
fields, for systems both having and or lacking number conservation in two
dimensions (2D). For the latter, with strong activity or nonequilibrium drive,
the system can synchronize, or be phase-ordered with various types of order,
e.g., quasi long range order (QLRO) or new kind of order weaker or stronger
than QLRO for sufficiently strong active flow-phase couplings. For the number
conserving case, the system can show QLRO or order weaker than QLRO, again for
sufficiently strong active flow-phase couplings. For other choices of the model
parameters, the system necessarily disorders in a manner similar to immobile
but active XY spins, or 2D Kardar-Parisi-Zhang surfaces.
