  We show that certain isomorphisms of (twisted) KR-groups that underlie
T-dualities of torus orientifold string theories have purely algebraic
analogues in terms of algebraic K-theory of real varieties and equivalences of
derived categories of (twisted) coherent sheaves. The most interesting
conclusion is a kind of Mukai duality in which the "dual abelian variety" to a
smooth projective genus-1 curve over R with no real points is (mildly)
noncommutative.
