In the coming years, the FAIR accelerator will come online at GSI, opening up a new window into the phase structure of QCD. Now is the time to develop a better understanding of what the CBM experiment may find, with particular attention to signals of new physics. Lattice simulations can provide us first-principles information about QCD at zero density, but at nonzero density a sign problem arises from non-Hermiticity of the Dirac operator, obstructing our progress. In this talk, we bring insights from the field of non-Hermitian physics to bear on high energy theory, helping us develop new analytical and numerical tools for QCD phase structure. We show how non-Hermiticity can give rise to a “moat regime” in the vicinity of a critical point. We discuss experimental signatures for these phenomena at FAIR.