What happens to the output of a beam splitter when you change the relative phase between two photons that enter from the two input ports?
In Hong-Ou-Mandel interference for a beamsplitter of the form, where I represent my outputs as $b^\dagger_1$ and $b^\dagger_2$:
$$ \begin{equation*} \left(\begin{array}{cc} \hat{b}^\dagger_1\\ \hat{b}^{\dagger}_2 \\ \end{array}\right) = \frac{1}{\sqrt{2}} \left(\begin{array}{cc} 1 & -1 \\ 1 & 1 \\ \end{array}\right) \left(\begin{array}{cc} a^\dagger_1 \\ a^\dagger_2 \\ \end{array}\right) \end{equation*} $$
which implies the inputs have the relation: $$ \begin{equation*} \left(\begin{array}{cc} \hat{a}^\dagger_1\\ \hat{a}^{\dagger}_2 \\ \end{array}\right) = \frac{1}{\sqrt{2}} \left(\begin{array}{cc} 1 & 1 \\ -1 & 1 \\ \end{array}\right) \left(\begin{array}{cc} b^\dagger_1 \\ b^\dagger_2 \\ \end{array}\right) \end{equation*} $$
with an input of $|1, 1\rangle = a^\dagger_1 a^\dagger_2 |0, 0\rangle = \frac{1}{\sqrt{2}}(b^\dagger_1+b^\dagger_2)\frac{1}{\sqrt{2}}(-b^\dagger_1+b^\dagger_2)= \frac{1}{2}(-b^\dagger_1 b^\dagger_1-b^\dagger_2b^\dagger_1 +b^\dagger_1 b^\dagger_2+b^\dagger_2 b^\dagger_2) = \frac{1}{2}(-b^\dagger_1 b^\dagger_1+b^\dagger_2 b^\dagger_2)$
This math, to me, suggests that the resultant "two-photon interference" is invariant to the relative phase between the two fields. That is, if I add a phase $e^{i \phi}$ to one of my $a^\dagger$ modes, it just gets carried through the whole process as a global phase, without producing interference:
$|\tilde{1}, 1\rangle = \left(a^\dagger_1 e^{i \theta}\right) a^\dagger_2 |0, 0\rangle = e^{i \theta}(-b^\dagger_1 b^\dagger_1+b^\dagger_2 b^\dagger_2)$
This phase doesn't change the fact that the photons $|1, 1\rangle$ states destructively interfere. I thought this fact is aligned with the general intuition that "photons don't have well-defined phases'' because generally pure Fock states will often lose any phase given to them unless a relative phase is created (for instance putting a Fock state in a Mach-Zehnder interferometer).
But this conclusion appears to be in contradiction with this paper, which says that adding a relative phase to the photon pair ends up changing the interference, allowing to flip between bunching and antibunching depending on the phase.
In this paper they say that you can think of the result as a sort of post-selected Mach-Zehnder interferometer. If the first photon is found in detector 1, it means the second photon acts like it is in a Mach-Zehnder interferometer, and consequently can be routed into either detector changing the relative phase between paths. To quote:
So what exactly is wrong about this previous treatment?