  The question of whether the holomorphic collinear singularities of graviton
amplitudes define a consistent chiral algebra has garnered much recent
attention. We analyse a version of this question for infinitesimal
perturbations around the self-dual sector of 4d Einstein gravity. The
singularities of tree amplitudes in such perturbations do form a consistent
chiral algebra, however at 1-loop its operator products are corrected by the
effective graviton vertex. We argue that this chiral algebra can be interpreted
as the universal holomorphic surface defect in the twistor uplift of self-dual
gravity, and show that the same correction is induced by an anomalous diagram
in the bulk-defect system. The 1-loop holomorphic collinear singularities do
not form a consistent chiral algebra. The failure of associativity can be
traced to the existence of a recently discovered gravitational anomaly on
twistor space. It can be restored by coupling to an unusual 4th-order
gravitational axion, which cancels the anomaly by a Green-Schwarz mechanism.
Alternatively, the anomaly vanishes in certain theories of self-dual gravity
coupled to matter, including in self-dual supergravity.
