We give a method to construct singular Lagrangian 3-torus fibrations over certain a priori given integral affine manifolds with singularities, which we call simple. The main result of this article is the proof that M. Gross' topological Calabi-Yau compactifications [7] can be made into symplectic compactifications. As an example, we obtain a pair of compact symplectic 6-manifolds together with Lagrangian fibrations whose underlying affine structures are dual. The symplectic manifolds obtained are homeomorphic to a smooth quintic Calabi-Yau 3-fold and its mirror.