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.
On the rationality of some real threefolds
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We study the rationality of some geometrically rational three-dimensional conic and quadric surface bundles, defined over the reals and more general real closed fields, for which the real locus is connected and the intermediate Jacobian obstructions to rationality vanish. We obtain both negative and positive results, using unramified cohomology and birational rigidity techniques, as well as concrete rationality constructions.
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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.