  B-fields over a groupoid with involution are defined as Real graded
Dixmier-Douady bundles. We use these to introduce the Real graded Brauer group
which constitutes the set of twistings for Atiyah's KR-functor in the category
of locally compact groupoids with involutions. We interpret this group in terms
of groupoid extensions and elements of some equivariant Cech cohomology theory.
The construction of the twisted KR-functor is outlined.
