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We report a global structural distortion in Sr$_2$IrO$_4$ using spatiallyresolved optical second and third harmonic generation rotational anisotropymeasurements. A symmetry lowering from an $I4_{1}/acd$ to $I4_{1}/a$ spacegroup is observed both above and below the N\'{e}el temperature that arisesfrom a staggered tetragonal distortion of the oxygen octahedra. By studying aneffective super-exchange Hamiltonian that accounts for this lowered symmetry,we find that perfect locking between the octahedral rotation and magneticmoment canting angles can persist even in the presence of large non-cubic localdistortions. Our results explain the origin of the forbidden Bragg peaksrecently observed in neutron diffraction experiments and reconcile theobservations of strong tetragonal distortion and perfect magneto-elasticlocking in Sr$_2$IrO$_4$.

A structural distortion induced magneto-elastic locking in Sr$_2$IrO$_4$ revealed through nonlinear optical harmonic generation