  In this article, we study mirror symmetry for pairs of singular Calabi--Yau
manifolds which are double covers of toric manifolds. Their period integrals
can be seen as certain `fractional' analogues of those of ordinary complete
intersections. This new structure can then be used to solve their
Riemann--Hilbert problems. The latter can then be used to answer definitively
questions about mirror symmetry for this class of Calabi--Yau manifolds.
