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The seminormalization of an algebraic variety $X$ is the biggest varietylinked to $X$ by a finite, birational and bijective morphism. In this paper weintroduce a variant of the seminormalization, suited for real algebraicvarieties, called the R-seminormalization. This object have a universalproperty of the same kind of the one of the seminormalization but related tothe real closed points of the variety. In a previous paper, the author studiedthe seminormalization of complex algebraic varieties using rational functionsthat extend continuously to the closed points for the Euclidean topology. Weadapt here some of those results to the R-seminormalization and we provideseveral examples. We also show that the R-seminormalization modifies thesingularities of a real variety by normalizing the purely complex points andseminormalizing the real ones.

A notion of seminormalization for real algebraic varieties