Recursive subhomogeneity of orbit-breaking subalgebras of ${\mathrm{C}}^{\ast }$-algebras associated to minimal homeomorphisms twisted by line bundles

In this paper, we construct a recursive subhomogeneous decomposition for the Cuntz–Pimsner algebras obtained from breaking the orbit of a minimal Hilbert $C(X)$-bimodule at a subset $Y\subset X$ with non-empty interior. This generalizes the known recursive subhomogeneous decomposition for orbit-breaking subalgebras of crossed products by minimal homeomorphisms.