Variétés réelles connexes non stablement rationnelles

Let $R$ be the field of real Puiseux series. It is a real closed field. We construct the first examples of smooth intersections of two quadrics in ${\mathbb{P}}_R^5$ and smooth cubic hypersurfaces in ${\mathbb{P}}_R^4$ which are not stably rational but for which the space $X(R)$ of $R$-points is semi-algebraically connected. The question of constructing such examples over the field of real numbers $\mathbb{R}$ remains open.