We reformulate twistor-string theory as a heterotic string based on a twisted
(0,2) model. The path integral localizes on holomorphic maps, while the (0,2)
moduli naturally correspond to the states of N=4 super Yang-Mills and conformal
supergravity under the Penrose transform. We show how the standard
twistor-string formulae of scattering amplitudes as integrals over the space of
curves in supertwistor space may be obtained from our model. The corresponding
string field theory gives rise to a twistor action for N=4 conformal
supergravity coupled to super Yang-Mills. The model helps to explain how the
twistor-strings of Witten and Berkovits are related and clarifies various
aspects of each of these models.Comment: 45 page