In split signature, global twistor constructions for conformally self-dual (SD) gravity and Yang-Mills construct solutions from twistor data that can be expressed in terms of free functions without gauge freedom. This is developed for asymptotically flat SD gravity to give a fully nonlinear encoding of the asymptotic gravitational data in terms of a real homogeneous generating function h on the real twistor space. The recently discovered Lw1+∞ celestial symmetries, when real, act locally as passive Poisson diffeomorphisms on the real twistor space. The twistor data, h, generates an imaginary such Poisson transformation that then generates the gravitational field by shifting the real slice of the twistor space. The twistor chiral sigma models, whose correlators yield the Einstein gravity tree-level S-matrix, are reformulated as theories of holomorphic discs in twistor space whose boundaries lie on the deformed real slice determined by h. The real Lw1+∞ symmetries act on the corresponding formula for the S-matrix geometrically with vanishing Noether currents, but imaginary generators yield graviton vertex operators that generate gravitons in the perturbative expansion. This talk is based on https://arxiv.org/abs/2212.10895.