  Witten has recently proposed a string theory in twistor space whose
D-instanton contributions are conjectured to compute N=4 super-Yang-Mills
scattering amplitudes. An alternative string theory in twistor space was then
proposed whose open string tree amplitudes reproduce the D-instanton
computations of maximal degree in Witten's model.
  In this paper, a cubic open string field theory action is constructed for
this alternative string in twistor space, and is shown to be invariant under
parity transformations which exchange MHV and googly amplitudes. Since the
string field theory action is gauge-invariant and reproduces the correct cubic
super-Yang-Mills interactions, it provides strong support for the conjecture
that the string theory correctly computes N-point super-Yang-Mills tree
amplitudes.
