We study AdS(7) vacua of massive type IIA string theory compactified on a 3-sphere with H-3 flux and anti-D6-branes. In such backgrounds, the anti-brane backreaction is known to generate a singularity in the H-3 energy density, whose interpretation has not been understood so far. We first consider supersymmetric solutions of this setup and give an analytic proof that the flux singularity is resolved there by a polarization of the anti-D6-branes into a D8-brane, which wraps a finite 2-sphere inside of the compact space. To this end, we compute the potential for a spherical probe D8-brane on top of a background with backreacting anti-D6-branes and show that it has a local maximum at zero radius and a local minimum at a finite radius of the 2-sphere. The polarization is triggered by a term in the potential due to the AdS curvature and does therefore not occur in non-compact setups where the 7d external spacetime is Minkowski. We furthermore find numerical evidence for the existence of non-supersymmetric solutions in our setup. This is supported by the observation that the general solution to the equations of motion has a continuous parameter that is suggestive of a modulus and appears to control supersymmetry breaking. Analyzing the polarization potential for the non-supersymmetric solutions, we find that the flux singularities are resolved there by brane polarization as well.