Alternatively, and thinking purely classically, if we do t -> -t, what happens to a perfect absorber, which takes in anything and reflects nothing impinging on it? It should spit out arbitrary things and reflect everything impinging on it. (Here I'm thinking of the Poynting vector for plane waves with their origin below the photon sphere and moving outwards vs just above the antihorizon and moving inwards -- the latter can't enter the WH).
(Maybe easier to think about if the BH absorption isn't total, like if we spin the black hole and have the plane waves (or their polarization) move against the rotation, and the WH is just time reversed -- the WH emits a bit of radiation which joins more outside the antihorizon, and the combination flies off to infinity, and not thinking too hard about the interior metric; although Wald deals with much of this in the text around eqn (12.4.18)).