We investigate the dispersions of anyon quasiparticles in the Kitaev honeycomb spin liquid perturbed by Γ and \({\Gamma }^{{\prime} }\) couplings in order to understand phase transitions into competing states through anyon gap-closing instabilities. We demonstrate how anyon gap closings provide a way to understand phase transitions into a plethora of previously identified competing phases—including zigzag, stripy, 120°, and incommensurate spiral phases—and are in agreement with numerical studies not only on the nature of the phases, but also on the specific critical values of Γ and \({\Gamma }^{{\prime} }\) couplings. Remarkably, when the antiferromagnetic Kitaev model is perturbed by a negative Γ interaction, we find that the single-vison and fermion gaps remain open while the gap of a magnon-like local boson vanishes. This points to a plausible candidate coexistence regime where long-range magnetic order emerges while fractionalized excitations inherited from the Kitaev spin liquid may persist. The magnetic order can be either stripy antiferromagnetic or incommensurate spiral, depending on the sign of \({\Gamma }^{{\prime} }\).
