For smooth geometrically rational real Fano threefolds with nonempty real locus, the absence of any deformation-equivalent real form with disconnected real locus forces R-rationality.
Smooth Fano 3-folds satisfying Condition (A)
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abstract
A smooth variety is said to satisfy Condition (A) if every finite abelian subgroup of its automorphism group has a fixed point. We classify smooth Fano 3-folds that satisfy Condition (A).
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A rationality criterion for real Fano threefolds
For smooth geometrically rational real Fano threefolds with nonempty real locus, the absence of any deformation-equivalent real form with disconnected real locus forces R-rationality.