Holomorphic splitting of a twistor patching matrix ($F=\tilde{H}{H}^{-1}$)

A bundle trivial on a twistor line has a global holomorphic bundle frame there. Comparing it with the given two patch bundle frames produces invertible matrices $H,\tilde{H}$, holomorphic on their respective affine patches, with $F=\tilde{H}{H}^{-1}$. The obstruction to varying such a bundle frame locally with space-time is $H^1({\mathbb{CP}}^1,\mathcal{O}{)}^{n^2}=0$, so the splitting factors can be chosen holomorphically in a neighbourhood. If two splittings differ by $H\mapsto Hg$ and $\tilde{H}\mapsto \tilde{H}g$, then $g$ is holomorphic on the compact complex projective line and hence independent of its projective coordinate. It is the space-time gauge freedom.