  We consider holomorphic Poisson-BF theory on twistor space. Classically, this
describes self-dual Einstein gravity on space-time, but at the quantum level it
is plagued by an anomaly. The anomaly corresponds to the fact that
integrability of the self-dual vacuum Einstein equations does not survive in
self-dual quantum gravity. We compute the anomaly polynomials in the Poisson-BF
theory, as well as in this theory coupled to a holomorphic BF theory on twistor
space describing self-dual Yang-Mills. We show that all anomalies may be
cancelled by further coupling to a twistor field representing a type of axion
on space-time. When the twistor anomalies are cancelled, all $n\geq4$-pt
amplitudes vanish and integrability is restored.
