Given a locally compact Hausdorff groupoid $\mathcal{G}$ with Haar system, the twistings of its $K$-groups are defined as Morita equivalence classes of graded Dixmier-Douady bundles, or equivalently, as equivalence classes of graded $\mathbb{S}^1$-central extensions of some groupoids Morita equivalent to $\mathcal{G}$. It is well known that these elements form an abelian group $\widehat{Br}(\mathcal{G})$, called the graded Brauer group of $\mathcal{G}$, which is isomorphic to $\check{H}^1(\mathcal{G}_\bullet,\mathbb{Z}_2) \oplus \check{H}^2(\mathcal{G}_\bullet,\mathbb{S}^1)$. In my talk I introduce the Real graded Brauer group $\widehat{BrR}(\mathcal{G})$ that constitutes the group of twistings for the twisted analog of Atiyah's $KR$-theory of locally compact Real groupoids (i.e. groupoids endowed with involutions). I will then give a cohomological formula of this group.