Let's consider interference between two photons. If we "rotate" one of the wavefunctions with a phase factor, this will affect the interference, and the result will depend on the phase factor. So, the interference between $|1\rangle$ and $|2\rangle$ is different than the interference between $|1\rangle$ and $e^{i\vartheta}|2\rangle$. This can be checked experimentally by using a phase shifter.

Let $\frac 1 {\sqrt 2}\left(|1\rangle|2\rangle + |2\rangle|1\rangle\right)$ be two photons. The same state can be represented in the Fock space as $\frac 1 {\sqrt 2}\left(|1'\rangle|2'\rangle + |2'\rangle|1'\rangle\right)$, where $|1'\rangle=e^{i\varphi}|1\rangle$, and $|2'\rangle=e^{-i\varphi}|2\rangle$, and $\varphi$ is an arbitrary overall phase.

Let's write the interference:

$$|1'\rangle+|2'\rangle = e^{i\varphi}|1\rangle + e^{-i\varphi}|2\rangle = e^{i\varphi}\left(|1\rangle + e^{-2i\varphi}|2\rangle\right).$$

This can't be equal to $|1\rangle+|2\rangle$, not even up to a phase factor.

Apparently, the interference between photons depends on the way we decompose the state from the Fock space: in terms of $|1\rangle,|2\rangle$ we obtain different interference than in terms of $|1'\rangle,|2'\rangle$. So, either the Hilbert space does not contain all the needed information, or I am missing something (which is by far more likely). Could you please explain me where I am wrong, and what is the correct description of interference using the Fock space?

P.S. Let me emphasize that I am fully aware that the Hilbert (and Fock in particular) space formalism has been tested successfully by so many experiments, and that most likely I am missing something. I just don't know what.