Can mobility induce orders in active XY spins on a substrate?

We elucidate that the nearly phase-ordered active XY spins in contact with a conserved, diffusing species on a substrate can be stable. For wide-ranging model parameters, it has stable uniform phases robust against noises with orientational order logarithmically stronger or weaker than in equilibrium, together with miniscule (i.e., hyperuniform) or giant number fluctuations, respectively, establishing a direct correspondence between the two. The scaling of both phase and density fluctuations in the stable phase-ordered states is nonuniversal: they depend on the nonlinear dynamical couplings. For other parameters, it has no stable uniformly ordered phase. Our model, a theory for active spinners, provides a minimal framework for wide-ranging systems, e.g., active superfluids on substrates, synchronization of oscillators, active carpets of cilia and bacterial flagella, active membranes and sandblasting.