  We develop a method for the simulation of scalar field theories with complex
actions which is local, simple to implement and can be used in any number of
space-time dimensions. For model systems satisfying the $\mathcal{PT}$ symmetry
condition $L^{*}(\phi)=L(-\phi)$, the complex weight problem is reduced to a
sign problem. The sign problem is eliminated completely for a large subclass of
these models; this class includes models within the $i\phi^{3}$ universality
class, and also models with nonzero chemical potential. Simulations of models
from this subclass show a rich set of behaviors. Propagators may exhibit damped
oscillations, indicating a clear violation of spectral positivity. Modulated
phases occur in some models, exhibiting striping and other pattern-forming
behaviors. These field theory models are connected to complex systems where
pattern formation occurs because of competition between interactions at two
different length scales.
