In solid physics, why can the wavefunction in real space be real-valued when only the $\Gamma(k=0)$ point is considered?

Question

Packages like the Quantum Espresso use this technique, but I cannot figure out the reason. The Bloch's theorum states that for periodic systems, we have $$\psi(\vec{r}) = e^{i\vec{k}\cdot\vec{r}} u(\vec{r})$$ For $\Gamma$ point, the first term is only one, but is it guaranteed that $u$ can be chosen as a real-valued function? The proof seems to only require it to be periodic.