Strong existence and uniqueness of the tilt-indexed Busemann process in the planar corner growth model

We show that the Busemann process indexed by tilts in the super-differential of the limit shape exists and is unique in the strong sense in the i.i.d.\ planar corner growth model. This means that every probability space that supports the field of i.i.d.~weights supports a copy of the process and any two realizations of the process are equal almost surely.