  Lattice field theories with complex actions are not easily studied using
conventional analytic or simulation methods. However, a large class of these
models are invariant under CT, where C is charge conjugation and T is time
reversal, including models with non-zero chemical potential. For Abelian models
in this class, lattice duality maps models with complex actions into dual
models with real actions. For extended regions of parameter space, calculable
for each model, duality resolves the sign problem for both analytic methods and
computer simulations. Explicit duality relations are given for models for spin
and gauge models based on Z(N) and U(1) symmetry groups. The dual forms are
generalizations of the Z(N) chiral clock model and the lattice
Frenkel-Kontorova model, respectively. From these equivalences, rich sets of
spatially-modulated phases are found in the strong-coupling region of the
original models.
