(2+1)-dimensional [(2+1)D] topological orders possess emergent ordinary symmetries given by a group ${\text{Aut}}_{br}\left(\mathcal{C}\right)$, which consists of the braided tensor autoequivalences of the modular tensor category $\mathcal{C}$ that describes the anyons. In this paper we discuss cases where ${\text{Aut}}_{br}\left(\mathcal{C}\right)$ has elements that neither permute anyons nor are associated to any symmetry fractionalization but still acts faithfully on the theory, which we refer to as soft symmetries. We point out that one can construct topological defects corresponding to such exotic symmetry actions by decorating with a certain class of gauged symmetry protected topological states that cannot be distinguished by their torus partition function. This gives a physical interpretation to work by Davydov on soft braided tensor autoequivalences. This has a number of important implications for the classification of gapped boundaries, noninvertible spontaneous symmetry breaking, and the general classification of symmetry-enriched topological phases of matter. We also demonstrate analogous phenomena in higher dimensions, such as $(3+1)\mathrm{D}$ gauge theory with gauge group given by the quaternion group $Q_8$.

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• Spontaneous symmetry breaking
• Topological order
• Gauge theory techniques