  We consider six dimensional N=1 space-time supersymmetric Type IIB
orientifolds with non-zero untwisted NS-NS sector B-field. The B-field is
quantized due to the requirement that the Type IIB spectrum be left-right
symmetric. The presence of the B-field results in rank reduction of both 99 and
55 open string sector gauge groups. We point out that in some of the models
with non-zero B-field there are extra tensor multiplets in the Z_2 twisted
closed string sector, and we explain their origin in a simple example. Also,
the 59 open string sector states come with a multiplicity that depends on the
B-field. These two facts are in accord with anomaly cancellation requirements.
We point out relations between various orientifolds with and without the
B-field, and also discuss the F-theory duals of these models.
