  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.
