Supersymmetric probe branes satisfy the κ-symmetry condition which ensures that the action is minimized with respect to both variations of the embedding and of the world-volume gauge field. We observe that the κ-symmetry condition for a D7-brane in an imaginary self-dual (ISD) background can be generalized yielding minimization of the action with respect to variations of the gauge field only but not of the embedding. This provides a new way to construct non-BPS solutions for D7-branes once the embedding extremizes the geometrical volume of the brane. We then apply this method to the Klebanov-Strassler background and find a new D7-7 brane configuration that realizes the spontaneous breaking of flavor chiral symmetry as evidenced by the Goldstone boson identified in the spectrum. This result generalizes our previous construction for the Klebanov-Witten model. We compare our setup with the Sakai-Sugimoto model and discuss possible applications to QCD-like physics.