Conventional ordering transitions, described by the Landau paradigm, are characterized by the symmetries broken at the critical point. Within the constrained manifold occurring at low temperatures in certain frustrated systems, unconventional transitions are possible that defy this type of description. While the critical point exists only in the limit where defects in the constraint are vanishingly rare, unconventional criticality can be observed throughout a broad region of the phase diagram. This work presents a formalism for incorporating the effects of such defects within the framework of scaling theory and the renormalization group, leading to universal results for the critical behavior. The theory is applied to two transitions occurring within a model of spin ice, and the results are confirmed using Monte Carlo simulations. Relevance to experiments, particularly in the spin-ice compounds, is discussed, along with implications for simulations of related transitions, such as the cubic dimer model and the O(3) sigma model with "hedgehog" suppression. DOI: 10.1103/PhysRevB.87.064414