A Torelli theorem for moduli spaces of principal bundles on curves defined over $\mathbb{R}$

Let $X$ be a geometrically irreducible smooth projective curve, of genus at least three, defined over the field of real numbers. Let $G$ be a connected reductive affine algebraic group, defined over $\mathbb{R}$, such that $G$ is nonabelian and has one simple factor. We prove that the isomorphism class of the moduli space of principal $G$–bundles on $X$ determine uniquely the isomorphism class of $X$.