The breakdown of a complete factorization of non-perturbative physics into soft and collinear matrix elements is a generic feature of Soft-Collinear Effective Theory (SCET) at subleading power. I will present a chiral effective theory for the resulting non-local matrix elements, obtained by matching multi-local SCET operators onto hadronic degrees of freedom with a local realization of chiral symmetry. After introducing the degrees of freedom and power counting of the theory, I will discuss its application to the soft-overlap contribution to $B \to \pi$ form factors, where it predicts a qualitatively different pattern of chiral corrections from hard-pion chiral perturbation theory, including a non-factorization of the pion-mass and recoil-energy dependence. Finally, I will discuss how the hadronic effective theory provides an infrared description of the physics underlying rapidity singularities.