  The superspace geometry relevant to the heterotic string is reviewed from the
point of view of the off-shell supermultiplet structure of $N=1,d=10$
supergravity. The anomaly-modified seven-form Bianchi identity is analysed at
order $\a'^3$ and shown to admit a complete solution. The corresponding $\a'^3$
deformation of the dimension-zero torsion tensor is derived and shown to obey
the appropriate cohomological constraint.
