  We prove a decomposition theorem for the nef cone of smooth fiber products
over curves, subject to the necessary condition that their N\'eron--Severi
space decomposes. We apply it to describe the nef cone of so-called Schoen
varieties, which are the higher dimensional analogues of the Calabi--Yau
threefolds constructed by Schoen. Schoen varieties give rise to Calabi--Yau
pairs, and in each dimension at least three, there exist Schoen varieties with
non-polyhedral nef cone. We prove the Kawamata--Morrison--Totaro Cone
Conjecture for the nef cones of Schoen varieties, which generalizes the work by
Grassi and Morrison.
