If we place ourselves over an algebraically closed base field $k$, the étale localization datum can be described like so: the elementary étalés are the étale morphisms and the coverings are the families of étale morphisms $\prn{U\Sub{i}\to V}\Sub{i\in I}$ such that $U\Sub{i}\prn{k}$ covers $V\prn{k}$. Replacing $k$ simply with a real closed field (for example $\RR$) we obtain the real étale localization datum. The real étale topos of a variety is the little topos that we have constructed. The “local” rings are the Henselian local rings with real closed residue field, the “local” morphisms are the ordinary local morphisms, and the functions associated to the localization datum are the Nash functions (or the analytic-algebraic ones). One remarkable particularity is the fact that the real étale topos is a topos of sheaves on a topological space. For more details, see (Coste & Roy, 1981); for the moment, only the case of affine varieties is treated explicitly.

To conclude, we can remark that — apart from geometrical terminology — the only aspect of $M$ that we have utilized is the fact that it is a small category having finite projective limits. We therefore have here an all-purpose formalism which could be useful in domains other than classical or real algebraic geometry.