F-theory is a construction in string theory which uses elliptic fibrations from algebraic geometry to build compactifications of the type IIB string which are not purely geometric. In this talk (based on joint work with J. McOrist and S. Sethi), we will describe a way to use F-theory to build compactifications of the heterotic string which are not purely geometric. The construction is based on a refinement of the traditional duality between F-theory and the heterotic string which relates it to ``Shioda--Inose structures'' in algebraic geometry. These non-geometric heterotic compactifications include ones previously constructed by Hellerman, McGreevy, and Williams using other techniques.