  We consider actions of complex algebraic groups $\mathbf{G}$ on complex
algebraic varieties $\mathbf{X}$, coming from actions of real forms $G$ of
$\mathbf{G}$ and $X$ of $\mathbf{X}$. We explore the links between the real
points of the complex GIT quotient $\mathbf{X /\!\!/ G}$ and the real GIT
quotient $X /\!\!/ G$ defined by Richardson and Slodowy. We prove that some
type of real points of $\mathbf{X /\!\!/ G}$ can be lifted to a quotient of the
form $X /\!\!/ G$ maybe after changing the real forms, and we link the number
of possible lifts to a co-homology set. We apply then the results to character
varieties, and study the particular case of the
$\mathrm{SL}_3(\mathbb{C})$-character variety for $\mathbb{Z}$.
