quantum mechanics - Introducing a phase, what changes?

This question is related to: Mach-Zehnder interferometer and the Fresnel-Arago laws

Let us say we have unpolarised wave taking the form: $$\psi=\psi_0 e^{i(kx-\omega t)+i\phi(t)}$$ Where $\phi$ varies randomly with time. If I split this wave into two and send it through e.g. a double slit, one of the beams will experience a phase change due to an optical path length difference. When we combine these two waves one will take the form: $$\psi=\psi_0 e^{i(kx-\omega t)+i\phi(t)}$$ But what about the other?

Their are 3 possibilities: $$\psi=\psi_0 e^{i(k(x+x_0)-\omega t)+i\phi(t)}$$ $$\psi=\psi_0 e^{i(kx-\omega (t+t_0))+i\phi(t+t_0)}$$ $$\psi=\psi_0 e^{i(k(x+x_0)-\omega (t+t_0))+i\phi(t+t_0)}$$
Where $x_0$ and $t_0$ are constants. Which of these 3 is correct and why?

Answer

The two waves are interfering after having followed different paths, so $x$ must be different between the two. But you are observing them at the same time $t$ which must be the same for the two waves. So answer 1 is the good one.