X raised to the power x?

We note that $x=1$ is a trivial real solution to the equation. The other real solution is given as follows:

$\large \begin{aligned} x^{\sqrt[3]{x^2}} & = \sqrt x^x \\ x^{x^\frac 23} & = x^{\frac x2} \\ \implies x^\frac 23 & = \frac x2 \\ x^{1-\frac 23} & = 2 \\ x^\frac 13 & = 2 \\ \implies x & = 8 \end{aligned}$

Therefore, the sum of all real solutions is $1+8 = \boxed{9}$ .