We prove Fermat’s Last Theorem over and for prime exponents in certain congruence classes modulo by using a combination of the modular method and Brauer–Manin obstructions explicitly given by quadratic reciprocity constraints. The reciprocity constraint used to treat the case of is a generalization to a real quadratic base field of the one used by Chen and Siksek. For the case of , this is insufficient, and we generalize a reciprocity constraint of Bennett, Chen, Dahmen, and Yazdani using Hilbert symbols from the rational field to certain real quadratic fields.