  We present a systematic way of generating F-theory models dual to
nonperturbative vacua (i.e., vacua with extra tensor multiplets) of heterotic
E8xE8 strings compactified on K3, using hypersurfaces in toric varieties. In
all cases, the Calabi-Yau is an elliptic fibration over a blow up of the
Hirzebruch surface F_n. We find that in most cases the fan of the base of the
elliptic fibration is visible in the dual polyhedron of the Calabi-Yau, and
that the extra tensor multiplets are represented as points corresponding to the
blow-ups of the F_n.
