We prove continuous symmetry breaking in three dimensions for a special class of disordered models described by the Nishimori line. The spins take values in a group such as $\mathbb{S}^1$, $SU(n)$ or $SO(n)$. Our proof is based on a theorem about group synchronization proved by Abbe, Massouli\'e, Montanari, Sly and Srivastava [AMM+18]. It also relies on a gauge transformation acting jointly on the disorder and the spin configurations due to Nishimori [Nis81, GHLDB85]. The proof does not use reflection positivity. The correlation inequalities of [MMSP78] imply symmetry breaking for the classical $XY$ model without disorder.
Comment: 22 pages. (Added 1. The case where a (quenched) magnetic field is applied at each vertex 2. Symmetry breaking of left-isoclinic rotations for classical O(4) model and 3. How to recover Long-range-order for classical XY model using [MMSP78])