Rigidity questions for real half-classical manifolds

Let $X$ be a noncommutative real compact algebraic manifold, in the sense that $C(X)=C^*(x_1,\ldots,x_N|x_i=x_i^*,f_\alpha(x_1,\ldots,x_N)=0)$, with $f_\alpha\in\mathbb R<x_1,\ldots,x_N>$. Associated to $X$ are its classical version $X^\times$, obtained via the relations $ab=ba$, and its half-classical version $X^*$, obtained via the relations $abc=cba$. We discuss here some general questions regarding the inclusions $X^\times\subset X^*\subset X$, and notably the comparison of the corresponding quantum isometry groups. Our main results concern the half-classical case, $X=X^*$, and more specifically, the case of the submanifolds $X\subset S^{N-1}_{\mathbb R,*}$.