  In this paper we construct twisted Real quasi-elliptic cohomology as the
twisted KR-theory of loop groupoids. The theory systematically incorporates
loop rotation and reflection. After establishing basic properties of the
theory, we construct twisted elliptic Pontryagin characters and, without
twists, Real analogues of the string power operation of quasi-elliptic
cohomology. We also explore the relation of the theory to the Tate curve.
