@Z.M The locale in Clausen-Scholze is clearly not the classical one.

@Z.M Here the map considered is from a Riemann surface to a point. "Affine opens proper over Y" is just analytic spectra of overconvergent rings. I am viewing analytic spaces as locally analytically ringed locales to talk about openness. Actually the way to make a Riemann surface an analytic space is exactly to glue from analytic spectra of overconvergent rings on closed discs, so these serve as "affine opens" in analog with scheme theory.