I would like to know the set $X=J^{-1}(\mathbb{R})$, where $J$ is Klein's $J$-invariant. Since $J$ is a modular function, it suffices to know the intersection of $X$ and the fundamental domain $\{\tau\in\mathbb{H}\mid |\tau|\ge 1,|\mathrm{Re}(\tau)| \le1/2\}$.
The special values of $J$ suggest that $J(\tau)>0$ for $\tau=i y$ with $y\ge 1$, but I cannot prove that it holds generally.
The $j$-invariant is real on the fundamental domain, if and only if $\tau$ lies on the boundary (both vertical lines and the circular arc) or if $\tau$ has zero real part (lies on the imaginary axis).
To prove this, note that on the vertical lines with integer and half-integer real values, the $h$-function is real, and the unit semicircle is equivalent uunder $\text{SL}_2(\Bbb Z)$ to one of these vertical lines, so $j$ is real where I said.
Also $j$ tends to $\infty$ and $-\infty$ as $\tau\to\infty$ on these vertical lines, so takes every real value, by the IVT, on the stated set. Now since the $j$-function is bijective on the "strict" fundamental region, then elsewhere the $j$-function is non-real.