We demonstrate that the one-loop exact subleading soft graviton theorem automatically follows from conservation of the BMS charges, provided that the hard and soft fluxes separately represent the extended BMS algebra at null infinity. This confirms that superrotations are genuine symmetries of the gravitational S-matrix beyond the semiclassical regime. In contrast with a previous proposal, the celestial stress tensor accounting for the one-loop corrections follows from the gravitational phase space analysis and does not require the addition of divergent counterterms. Moreover, we show that the symplectic form on the radiative phase space factorises into hard and soft sectors, and that the resulting canonical generators precisely coincide with the correct BMS fluxes.
