Determination of the properties of high-density QCD is important, but difficult due to a sign problem. Methods based on PT symmetry offer new ideas for attacking this problem and connect it to problems in other areas of physics. Using both simulations and analytic methods, a model field theory with a sign problem is shown to have inhomogeneous, patterned phases. The formation of patterned phases can be understood as a consequence of a Lifshitz instability. Analytic results from a liquid-gas model related to dense QCD with heavy quarks show a similar phase diagram. Recent results from a real-space renormalization group analysis also show exotic phases in Z(N) lattice field theories at finite density.