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We use Menke's JSJ-type decomposition theorem for symplectic fillings toreduce the classification of strong and exact symplectic fillings of virtuallyovertwisted torus bundles to the same problem for tight lens spaces. Forvirtually overtwisted structures on elliptic or parabolic torus bundles, thisgives a complete classification. For virtually overtwisted structures onhyperbolic torus bundles, we show that every strong or exact filling arisesfrom a filling of a tight lens space via round symplectic 1-handle attachment,and we give a condition under which distinct tight lens space fillings yieldthe same torus bundle filling.

On symplectic fillings of virtually overtwisted torus bundles