  Many scalar field theory models with complex actions are invariant under the
antilinear ($PT$) symmetry operation $L^{\ast}(-\chi)=L(\chi)$. Models in this
class include the $i\phi^{3}$ model, the Bose gas at finite density and
Polyakov loop spin models at finite density. This symmetry may be used to
obtain a dual representation where weights in the functional integral are real
but not necessarily positive. For a subclass of models satisfying a dual
positive weight condition, the partition function is manifestly positive. The
sign problem is eliminated; such models are easily simulated by a simple local
algorithm in any number of dimensions. Simulations of models in this subclass
show a rich set of behaviors. Propagators may exhibit damped oscillations,
indicating a clear violation of spectral positivity. Pattern formation may also
occur, with both stripe and bubble morphologies possible. The existence of a
positive representation is constrained by Lee-Yang zeros: a positive
representation cannot exist everywhere in the neighborhood of such a zero.
Simulation results raise the possibility that pattern-forming behavior may
occur in finite density QCD in the vicinity of the critical line.
