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We consider Real bundle gerbes on manifolds equipped with an involution andprove that they are classified by their Real Dixmier-Douady class inGrothendieck's equivariant sheaf cohomology. We show that the Grothendieckgroup of Real bundle gerbe modules is isomorphic to twisted KR-theory for atorsion Real Dixmier-Douady class. Using these modules as building blocks, weintroduce geometric cycles for twisted KR-homology and prove that they generatea real-oriented generalised homology theory dual to twisted KR-theory for Realclosed manifolds, and more generally for Real finite CW-complexes, for any RealDixmier-Douady class. This is achieved by defining an explicit naturaltransformation to analytic twisted KR-homology and proving that it is anisomorphism. Our model both refines and extends previous results by Wang andBaum-Carey-Wang to the Real setting. Our constructions further provide a newframework for the classification of orientifolds in string theory, providingprecise conditions for orientifold lifts of H-fluxes and for orientifoldprojections of open string states.

Real bundle gerbes, orientifolds and twisted KR-homology