  The existence of interacting higher-spin theories is tightly constrained by
many no-go theorems. In this paper, we construct a chiral, higher-spin
generalization of Yang-Mills theory in flat space which avoids these no-go
theorems and has non-trivial tree-level scattering amplitudes with some
higher-spin external legs. The fields and action are complex, so the theory is
non-unitary and parity-violating, yet we find surprisingly compact formulae for
all-multiplicity tree-level scattering amplitudes in the maximal helicity
violating (MHV) sector, where the two negative helicity particles have
identical but arbitrary spin. This is possible because the theory admits a
perturbative expansion around its self-dual sector. Using twistor theory, we
prove the classical integrability of this self-dual sector and show that it can
be described on spacetime by an infinite tower of interacting massless scalar
fields. We also give a twistor construction of the full theory, and use it to
derive the formula for the MHV amplitude.
