  The goal of this article is to show that Hitchin representations over real
closed field extensions $\mathbb{F}$ of $\mathbb{R}$ are exactly those
representations from the fundamental group of a closed surface into
$\textrm{PSL}(n,\mathbb{F})$ that are conjugate to $\mathbb{F}$-positive
representations, i.e. representations that admit an equivariant limit map from
the set of fixed points in the boundary of the universal cover of the surface
into the set of full flags in $\mathbb{F}^n$ satisfying specific positivity
properties. They form semi-algebraically connected components of the space of
all representations that consist entirely of injective representations with
discrete image.
