Let $f(z)$ be a function meromorphic in a simply connected convex domain $D$ (subset of the complex plane with positive area or the whole complex plane) where $z$ is a complex number.
Are there such functions $f(z)$ where $\Re(f(z))$ is periodic in the domain (no periods larger than the domain please :p ) but $f(z)$ is not periodic? (if $D\subset \mathbb C$ it is clear that $f(z)$ is not periodic but $\Re(f(z))$ might still be for some shapes of $D$).
In particular the case when $D = \mathbb C$ is interesting. (in other words $f(z)$ meromorphic over $\mathbb C$)
I guess it is a similar question to ask about $\Im$ , $\operatorname{Arg}$ or $|\cdot|$ instead of $\Re$.
I read about double periodic functions and Cauchy-Riemann equations but I still don't know. I can't find such a function in the literature ( i mean the one i search here , i don't mean i can't find a double periodic one in the literature of course ) and I don't know how to construct them or even if they exist.