  The possibility of spontaneous breaking of CP symmetry by the expectation
values of orbifold moduli is investigated with particular reference to $CP$
violating phases in soft supersymmetry breaking terms. The effect of different
mechanisms for stabilizing the dilaton and the form of the non-perturbative
superpotential on the existence and size of these phases is studied.
Non-perturbative superpotentials involving the absolute modular invariant
$j(T)$, such as may arise from F-theory compactifications, are considered.
