Weakly breaking a continuous symmetry in an ordered phase can convert its Goldstone modes into topological collective bands. We establish this mechanism in a neutral dipole condensate whose conserving parent supports two independent Goldstone phase modes. Weak real flavor mixing pins the phases and selects a chiral condensate. The resulting gyroscopic coupling acts on their winding polarization texture, producing a pseudo-Goldstone sector containing a topological lowest band. Restoring dipole conservation forces the two modes to meet at zero frequency, where their separate band Chern numbers cease to be defined. The same parent dipole charges define a tensor current whose circular response permits model-assisted reconstruction of low-energy embedded local Berry curvature in the benchmark regime. Analytical stability bounds and a large-occupation limit provide a controlled realization of this mechanism.