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The quotient $h=f/g$ is a holomorphic function because the denominator has no zeros, and $|h|=1$. Write $h=u+iv$. The Cauchy-Riemann equations say $u_y=-v_x$ and $v_y=u_x$. Differentiating $u^2+v^2=1$ in both real directions givesThe coefficient determinant is $-(u^2+v^2)=-1$, so $u_x=v_x=0$, and the Cauchy-Riemann equations also give $u_y=v_y=0$. Thus $h$ is locally constant. Since a domain is connected, it is constant throughout $\Omega$. Its constant value has modulus one and can be written $e^{i\alpha }$ with $\alpha \in \mathbb{R}$. Therefore $f=e^{i\alpha }g$ throughout $\Omega$.

This is the constant-modulus holomorphic function principle. Connectedness is essential to a single global phase: on two disjoint discs, taking $g=1$ and choosing $f=1$ on one disc and $f=-1$ on the other gives equal moduli without one common phase.