  We consider chiral gauge theories defined over a four-dimensional spacetime
manifold with a Cartesian product structure for at least one compact spatial
dimension. For a simple setup, we calculate the effective gauge field action by
integrating out the chiral fermions, while maintaining gauge invariance. Due to
a combination of infrared and ultraviolet effects, there appears a CPT-odd term
in the effective gauge field action. This CPT anomaly could occur in chiral
gauge theories relevant to elementary particle physics, provided the spacetime
manifold has the appropriate topology. Two possible applications for cosmology
are discussed.
