The long-run behavior of multi-agent online learning -- and, in particular, no-regret learning -- is relatively well-understood in potential games, where players have common interests. By contrast, in general harmonic games -- the strategic complement of potential games, where players have competing interests -- very little is known outside the narrow subclass of 2 -player zero-sum games with a fully-mixed equilibrium. As a first result, we show that the continuous-time dynamics of FTRL are Poincaré recurrent, i.e., they return arbitrarily close to their starting point infinitely often, and hence fail to converge. In discrete time, the standard, "vanilla" implementation of FTRL may lead to even worse outcomes, eventually trapping the players in a perpetual cycle of best-responses. However, if FTRL is augmented with a suitable extrapolation step -- which includes as special cases the optimistic and mirror-prox variants of FTRL -- we show that learning converges to a Nash equilibrium from any initial condition, and all players are guaranteed at most \mathcal{O}(1) regret.