The stable Galois correspondence for real closed fields

In previous work, the authors constructed and studied a lift of the Galois correspondence to stable homotopy categories. In particular, if $L/k$ is a finite Galois extension of fields with Galois group $G$, there is a functor $c_{L/k}^*$ from the $G$-equivariant stable homotopy category to the stable motivic homotopy category over $k$ such that $c_{L/k}^*(G/H_+) = Spec(L^H)_+$. We proved that when $k$ is a real closed field and $L=k[i]$, the restriction of $c_{L/k}^*$ to the $\eta$-complete subcategory is full and faithful. Here we "uncomplete" this theorem so that it applies to $c_{L/k}^*$ itself. Our main tools are Bachmann's theorem on the $(2,\eta)$-periodic stable motivic homotopy category and an isomorphism range for the map on bigraded stable stems induced by $C_2$-equivariant Betti realization.