  Finite-density QCD and many other field theories with sign problems have a
$\mathcal{PT}$-type symmetry. After a brief introduction to
$\mathcal{PT}$-symmetric field theories, a real dual representation for
$\mathcal{PT}$-symmetric scalar field theories with complex actions is derived.
We show that $\mathcal{PT}$-symmetric field theories can exhibit exotic
behavior, including sinusoidally modulated propagators, disorder lines, and
spatially inhomogeneous pattern-forming phases. We discuss the interplay of
duality, $\mathcal{PT}$-symmetry and pattern formation using a $\phi^4$ model
and $Z(N)$ spin model with sign problems as examples. These behaviors may occur
in finite-density QCD and related models.
