In this paper we extend previous work on calculating
massless boundary Ramond sector spectra of open strings
to include cases with nonzero flat $B$ fields.
In such cases, D-branes are no longer well-modeled precisely
by sheaves, but rather they are replaced by `twisted' sheaves,
reflecting the fact that gauge transformations of the $B$ field
act as affine translations of the Chan-Paton factors.
As in previous work, we find that the massless boundary Ramond
sector states are counted by Ext groups -- this time,
Ext groups of twisted sheaves. As before, the computation of
BRST cohomology relies on physically realizing some spectral
sequences. Subtleties that cropped up in previous work also appear
here.