  In split or Kleinian signature, twistor constructions parametrize solutions
to both gauge and gravity self-duality (SD) equation from twistor data that can
be expressed in terms of free smooth data without gauge freedom. Here the
corresponding constructions are given for asymptotically flat SD gravity
providing a fully nonlinear encoding of the asymptotic gravitational data in
terms of a real homogeneous generating function $h$ on the real twistor space.
Geometrically $h$ determines a nonlinear deformation of the location of the
real twistor space $\RP^3$ inside the complex twistor space $\CP^3$.
  This presentation gives an optimal presentation of Strominger's recently
discovered $Lw_{1+\infty}$ celestial symmetries. These, when real, act locally
as passive Poisson diffeomorphisms on the real twistor space. However, when
imaginary, such Poisson transformations are active symmetries, and generate
changes to the gravitational field by deforming the location of the real slice
of the twistor space.
  Gravity amplitudes for the full, non-self-dual Einstein gravity, arise as
correlators of a chiral twistor sigma model. This is reformulated for split
signature as a theory of holomorphic discs in twistor space whose boundaries
lie on the deformed real slice determined by $h$. Real $Lw_{1+\infty}$
symmetries act oas gauge symmetries, but imaginary generators yield graviton
vertex operators that generate gravitons in the perturbative expansion.
  A generating function for the all plus 1-loop amplitude, an analogous
framework for Yang-Mills, possible interpretations in Lorentz signature and
similar open string formulations of twistor and ambitwistor strings in 4d in
split signature, are briefly discussed.
