Why do we use superposition instead of tensor product in interferometer? In the description of a neutron interferometer here, it says:

In an interferometer the incident beam is split into two (or more) separate beams. The beams travel along different paths where they are exposed to different potentials (which results in different phases). At some point the beams are brought together again and allowed to interfere. The resulting beam is the superposition of the separated beams:
  $$ \psi  = \psi_I + \psi_{II} $$

I am interested in why the total wavefunction is not written as $\psi=\psi_I\otimes \psi_{II}$? Because when they are separated, they should be considered in two physical systems and we should use tensor product to describe them, even they are later combined together, right?
 A: In a nutshell: these are not two different systems, but the probability amplitudes of the two different states of the same system. 
I do agree that simple discussions of two interfering waves (electromagnetic or particle waves) are in practice only complicating the matter, as opposed to thinking of a single wave in a multiply connected geometry. They however make the math manageable.
A: The confusion comes from "and allowed to interfere". Superposition is not interaction, and they are describing a superposition of two beams of neutrons.
It is  similar to the superposition of two laser beams split from the same original, which show interference fringes due to the superposition of the two beams, and it is well known that photons do not interact except in high orders with very low probabilities.
The whole neutron interferometer is considered one physical system where the neutrons are not interacting with each other  , it is only the potentials that change in too complicated a way to really solve the total system wavefunction, but the addition of the two partial ones at the end  is a good approximation.
Tensor products would have been used in the density matrix formalism where individual neutrons are considered. The psis in your quote are the beam psis , not the individual neutron ones.
