  We introduce the wrapped Donaldson-Fukaya category of a (generalized)
semi-toric SYZ fibration with Lagrangian section satisfying a tameness
condition at infinity. Examples include the Gross fibration on the complement
of an anti-canonical divisor in a toric Calabi-Yau 3-fold. We compute the
wrapped Floer cohomology of a Lagrangian section and find that it is the
algebra of functions on the Hori-Vafa mirror. The latter result is the key step
in proving homological mirror symmetry for this case. The techniques developed
here allow the construction in general of the wrapped Fukaya category on an
open Calabi-Yau manifold carrying an SYZ fibration with nice behavior at
infinity. We discuss the relation of this to the algebraic vs analytic aspects
of mirror symmetry.
