Unfortunate for my purposes, but good to know!

The space $R_{\mathbb{R}}(X)$ has points consisting of maps Spec $\mathbb{C} \rightarrow X$ over $\Spec \mathbb{R}$. So complex conjugation induces an involution. The natural target for $R_{\mathbb{R}}$ is $\mathbb{Z}/2$-spaces, i.e., spaces equipped with an action of $\mathbb{Z}/2$.

Another reference for the Postnikov tower / Serre spectral sequence approach is Mosher and Tangora, Cohomology operations and applications in homotopy theory.

Because of Bob Bruner's computer calculations, I know what the answers ought to be. But I'm looking for a conceptual way of obtaining the answers.