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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 somefundamental properties of this notion. As an illustration, for any $n \geq 1$,we construct the first example of a quasi-projective real surface whosemutually non-isomorphic real forms admit a moduli of dimension at least $n$,parametrised by the real points of an affine $n$-space. Expanding on theseconstructions, we can give quasi-projective real varieties of any dimensionwhose algebraic moduli of the non-isomorphic real forms has arbitrarilypositive dimension.

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