We provide a description of adequate categorical data to give a Turaev–Viro type state-sum construct of invariants of 3-manifolds with a system of defects, generalizing the Dijkgraaf–Witten type invariants of our earlier work. We term the defects in our construction defects-with-structure because algebraic data associated to them is in general richer than a module category over the spherical fusion category from which the theory is constructed when no defect is present.