Relative forms of real algebraic varieties and examples of quasi-projective surfaces with algebraic moduli of real forms

We propose a framework to give a precise meaning to the intuitive notion of "family of real forms of a variety parametrised by a variety" and study some fundamental properties of this notion. As an illustration, for any $n \geq 1$, we construct the first example of a quasi-projective real surface whose mutually non-isomorphic real forms admit a moduli of dimension at least $n$, parametrised by the real points of an affine $n$-space. Expanding on these constructions, we can give quasi-projective real varieties of any dimension whose algebraic moduli of the non-isomorphic real forms has arbitrarily positive dimension.
Comment: The article has a new title, since it now also discusses the concepts of real functors, relative real forms and the tautological complete relative form. This gives a clearer meaning to the notion of "family of real forms of a variety parametrised by a variety". The example of the quasi-projective surface with non-isomorphic real forms parametrised by affine n-space is recast in this new framework