Atiyah's $K$-theory with Reality produces a $\mathbb{Z}/2$-spectrum $KR$. But I am stuck as I don't really know what the values of this equivariant spectrum should be on representations.
Any representation of $\mathbb{Z}/2$ is, up to isomorphism, $\mathbb{R}^p\oplus i \mathbb{R}^q$ with the obvious involution. Since $KR$-theory is represented by $BU \times \mathbb{Z}$, I think that the values at least in $\mathbb{C}^n$ should be $$KR(\mathbb{R}^n\oplus i \mathbb{R}^n):= BU \times \mathbb{Z},$$ and the structure maps for the spheres $S^{\mathbb{C}} = S^{1,1}$ should be the adjoint map of the weak equivalence $$BU \times \mathbb{Z} \to \Omega^{1,1}(BU \times \mathbb{Z})$$ given by (equivariant) Bott Periodicity.
Question. But what happens with the rest of representations, what is $KR(\mathbb{R}^p\oplus i \mathbb{R}^q)$ for general $p,q$? And what are the structure maps $$S^{p,q} \wedge KR(\mathbb{R}^{p'}\oplus i \mathbb{R}^{q'}) \to KR(\mathbb{R}^{p+p'}\oplus i \mathbb{R}^{q+q'}) $$