Solution
= Solution
For $z\ne-1$, multiply numerator and denominator by $\overline z+1$. The real part is
$$
\operatorname{Re}\frac{z-1}{z+1}=\frac{|z|^2-1}{|z+1|^2}.
$$
Thus its zero locus is
$$
\boxed{z=e^{i\theta},\qquad \theta\not\equiv\pi\pmod{2\pi}.}
$$
It is the <unit circle> with the point $-1$ removed. That point cannot be restored by the zero numerator obtained after clearing a denominator: the original expression is undefined there.