  We construct a general class of Calabi--Yau threefolds from fiber products of
rational elliptic surfaces with section, generalizing a construction of Schoen
to include all Kodaira fiber types. The resulting threefolds each have two
elliptic fibrations with section over rational elliptic surfaces and blowups
thereof. These elliptic fibrations generally have nonzero Mordell--Weil rank.
Each of the elliptic fibrations has a physical interpretation in terms of a
six-dimensional F-theory model with one or more non-Higgsable abelian gauge
fields. Many of the models in this class have mild singularities that do not
admit a Calabi--Yau resolution; this does not seem to compromise the physical
integrity of the theory and can be associated in some cases with massless
hypermultiplets localized at the singular loci. In some of these constructions,
however, we find examples of abelian gauge fields that cannot be "unHiggsed" to
a nonabelian gauge field without producing unphysical singularities that cannot
be resolved. The models studied here can also be used to exhibit T-duality for
a class of little string theories.
