Interaction of molecular SF6- anions with intense near-IR laser pulses is found to produce cationic fragments by nonsequential multiple detachment. Dissociative ionization channels that lie more than 20 eV above the threshold energy for double detachment are observed. Mass-resolved product yields are presented and analyzed as a function of the femtosecond laser pulse energy, pulse shape, and polarization ellipticity. The observed strong suppression of multiple detachment by pre-pulses, induced with negative third-order dispersion of the transform-limited fs laser pulse, is interpreted as suppression of a nonsequential process by early single detachment. However, in contrast to the relatively simple picture of a rescattering mechanism characterized by acute sensitivity to polarization ellipticity that dominates double ionization of neutral species and was reported for the atomic F- anion, multiple detachment of the molecular anion is found to exhibit only mild ellipticity dependence.