A realistic example of a non identifiable model? Given the definition of statistical model identifiability : identifiable iff $P_\theta = P_{\theta'} \implies \theta = \theta' ~~ \forall \theta, \theta' \in \Theta$ we can see that, for example, $X \sim N(\frac{\theta_1}{\theta_2},1)$ would not be identifiable.
But can you make a simple, intuitive real-world example of a case when one would meet with an unidentifiable model ?

In quantum mechanics appears many non-identifiable models in the problem of phase estimation. For example, when you try to estimate a phase $\theta \in [0, 2\pi)$ of a qubit, the $X \sim Ber(p(\theta))$, where $p(\theta) = \frac{1}{2}\left[1 + \sin(\theta) \right]$ which is $\pi$-periodic.
