Symplectic torus bundles ξ:T²→ E→ B are classified by the second cohomology group of B with local coefficients H₁(T²). For B a compact, orientable surface, the main theorem of this paper gives a necessary and sufficient condition on the cohomology class corresponding to ξ for E to admit a symplectic structure compatible with the symplectic bundle structure of ξ: namely, that it be a torsion class. The proof is based on a group-extension-theoretic construction of J. Huebschmann, 1981. A key ingredient is the notion of fibrewise-localization.