When is a logarithm of the wavefunction well-defined?

It is sometimes convenient to write the wavefunction as
$$
Psi(x,t)~=~ e^{Phi(x,t)}
$$
and then work with $Phi$ instead. This is particularly sensible in the context of the WKB approximation, where $Phi$ is nice a smooth and slowly changing.

But, does $Phi$ exist in general? There’s a reason it might not. Let’s say that $Psi(x_0,t_0) = 0$. It could be that, near $(x,t) = (x_0,t_0)$, the wavefunction looks like

$$
Psi(x,t) = e^{i tan^{-1} frac{x-x_0}{t-t_0}}
text.
$$

(When I write $tan^{-1}$, I mean something like the C/python "atan2" function. It’s just the angle formed by the vector $(x-x_0,t-t_0)$. I hope that’s clear.)

So, this wavefunction is smooth and well-behaved at $(x_0,t_0)$, but the imaginary part of its logarithm can’t be consistently defined. I guess one could say that $Phi(x,t)$ has a branch point.

Obviously this can’t happen if there wavefunction never vanishes. The only place where I know I can guarantee $Psi ne 0$ is the ground state of a bosonic system. Are there other situations in which $Phi$ can be guaranteed to make sense?

(For that matter… does $Phi$ have a name?)