On some finiteness results in real \'etale cohomology

We show that for quasi-compact quasi-separated schemes of finite dimension, the constructibility condition in real \'etale cohomology agrees with a notion of constructibility arising naturally from topology. As application we prove that the derived direct image functor preserves constructibility under some assumptions, and compute the Grothendieck group of the constructible rational stable motivic homotopy category for reasonable schemes. We prove the generic base change property for constructible real \'etale sheaves, and deduce the same property for rational motivic spectra and $b$-sheaves.