  We use Menke's JSJ-type decomposition theorem for symplectic fillings to
reduce the classification of strong and exact symplectic fillings of virtually
overtwisted torus bundles to the same problem for tight lens spaces. For
virtually overtwisted structures on elliptic or parabolic torus bundles, this
gives a complete classification. For virtually overtwisted structures on
hyperbolic torus bundles, we show that every strong or exact filling arises
from 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 yield
the same torus bundle filling.
