  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 $X\subset S^{N-1}_{\mathbb R,*}$.
