  The construction of perturbative quantities on non-linear backgrounds leads
to the possibility of incorporating strong field effects in perturbation
theory. We continue a programme to construct QFT observables on self-dual
backgrounds. The approach works with asymptotic data for fields defined at null
infinity $\mathscr{I}$, extending earlier work on Yang-Mills amplitudes on
self-dual backgrounds to form factors and incorporating supersymmetry. Since
our analysis is based on reconstruction from data at null infinity, it
naturally ties into work on celestial and twisted holography. We study form
factors both in pure Yang-Mills and their supersymmetric counterparts in
$\mathcal{N}=4$ SYM, giving a full treatment of $\mathcal{N}=4$
super-Yang-Mills at null infinity and their self-dual nonlinear backgrounds. We
obtain tree-level MHV form factors around these backgrounds using new formulae
for lifting operators to twistor space leading to simple dressings of the
corresponding form factors around the vacuum. We give brief indications on how
to go beyond the MHV sector by introducing dressed versions of the MHV diagram
propagator. We discuss generating functionals of the MHV all plus 1-loop
amplitude in this context together with its various dual conformal
representations.
