In this article we study the transitivity of the group of automorphisms of
real algebraic surfaces. We characterize real algebraic surfaces with very
transitive automorphism groups. We give applications to the classification of
real algebraic models of compact surfaces: these applications yield new insight
into the geometry of the real locus, proving several surprising facts on this
geometry. This geometry can be thought of as a half-way point between the
biregular and birational geometries.Comment: Compositio Mathematica (2010) To appea
