  In his twistor-particle programme of the 1970's, Roger Penrose introduced a
representation of the massive particle phase space in terms of a pair of
twistors subject to an internal symmetry group. Here we use this representation
to introduce a chiral string whose target is a complexification of this space,
extended so as to incorporate supersymmetry. We show that the gauge anomalies
associated to the internal symmetry group vanish only for maximal
supersymmetry, and that correlators in these string models describe amplitudes
involving massive particles with manifest supersymmetry. The models and
amplitude formulae exhibit a double copy structure from gauge theory on the
Coulomb branch to gravity, although the graviton remains massless. The formulae
are closely related to those obtained earlier by the authors expressed in terms
of the polarised scattering equations.
