Exact mathbb{Z}₂ electromagnetic duality of mathbb{Z}₂ toric code is non-Clifford

The 2D mathbb{Z}₂ toric code admits a global symmetry exchanging electric and magnetic quasiparticles, known as electromagnetic duality. Known realizations include lattice translation symmetry, an exact mathbb{Z}₄ symmetry generated by a Clifford circuit, and an exact mathbb{Z}₂ symmetry generated by a non-Clifford circuit. We show that a Clifford electromagnetic duality cannot realize an exact internal mathbb{Z}₂ symmetry. This is proved rigorously for symmetries with coarse translation invariance by l lattice units for generic odd l. Therefore an exact internal mathbb{Z}₂ electromagnetic duality must be non-Clifford, whereas generic internal Clifford realization necessarily has mathbb{Z}_2^m algebra with mge 2. Our result suggests an unexpected connection between the algebra of exact electromagnetic duality and Clifford hierarchy of circuits.