We investigate nonreciprocal XY (NRXY) models defined on two-dimensional lattices in which the coupling strength of a spin with its neighbors varies with their position in the frame defined by the current spin orientation. As expected from the seminal work of Dadhichi et al., [Nonmutual torques and the unimportance of motility for long-range order in two-dimensional flocks, Phys. Rev. E 101, 052601 (2020)], we first show that nonreciprocity is akin to a self-propulsion: we derive a mean-field continuous theory identical to that of constant-density flocks. Like the latter, NRXY models exhibit a long-range ordered phase that is metastable to the nucleation of topological defects, and their asymptotic state is a dynamic foam of asters. We then show that in the metastable ordered phase, the lattice always induces anisotropy on large scales, pinning the direction of order and imposing a finite correlation length. Crucially, we demonstrate that this anisotropy is inescapable since, even when not explicitly present in the interaction rule, it is generated by fluctuations. In short, the ordered phase of lattice NRXY models is that of active clock models.

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• Flocking
• Dry active matter
• Living matter & active matter