Consider four points, $\sigma_i$ i=1,2,3,4 on the line $\mathrm{Im}(z) = 0$ in the complex plane $\mathbb{C}$. Does it exist a rational function of these four points which is $\mathrm{SL}(2,\mathbb{R})$ invariant, but not $\mathrm{SL}(2,\mathbb{C})$ invariant?

For sure, you can build an invariant (not rational) with this property: for instance suppose that the geodesic $\gamma$ (with respect to the natural hyperbolic metric of the halfplane) joining $1$ and $3$ with the geodesic $\delta$ joining $2$ and $4$ intersect at a point $P$. Choose an $\mathrm{SL}(2,\mathbb{R})$ element so that $P$ is sent to $i$ and $\sigma_1$ to $0$. It follows that $\sigma_4 = - 1/\sigma_2$. Since the group $\mathrm{SL}(2,\mathbb{R})$ has now been completely fixed, the location of $\sigma_2$ in this frame is an invariant.

Note however, that the cross ratio of these four points, computed in this frame, is a function quadratic in $\sigma_2$ therefore can not distinguish wether $\sigma_2$ is greater or smaller than 0! If we go to the Poincarè disk model, the two cases are mapped to each other by an inversion which is not an element of $\mathrm{Aut}(D)$, but it is an element of $\mathrm{SL}(2,\mathbb{C})$ therefore the cross-ratio cannot distinguish the two cases, which are however not equivalent under any $\mathrm{SL}(2,\mathbb{R})$ transformation!

The question is: is there a rational function of $\sigma_i$ that does the job?