  Systems with non-reciprocal interactions generically display time-dependent
states. These are routinely observed in finite systems, from neuroscience to
active matter, in which globally ordered oscillations exist. However, the
stability of these uniform non-reciprocal phases in noisy spatially-extended
systems, their fate in the thermodynamic limit, and the criticality of the
corresponding phase transitions are not fully understood. Here, we address
these questions by introducing a non-reciprocal generalization of the Ising
model and study its critical behavior by means of numerical and analytical
approaches. While the mean-field equations predict three stable homogeneous
phases (disordered, ordered and a time-dependent swap phase), our large scale
numerical simulations reveal a more complex picture. Static order is destroyed
in any finite dimension due to the growth of rare droplets. The swap phase is
destroyed by fluctuations in two dimensions through the proliferation of
discrete spiral defects, but stabilized in three dimensions where
non-reciprocity changes the universality class from Ising to XY giving rise to
robust spatially-distributed clocks.
