(This is essentially circular polarised light. Because if the two beams going to detector zero werent alternating between horizontal and vertical polarised light each cycle and instead were just two same plane polarised beams as some pretend, then the two phase shifted interference patterns and diffraction patterns observed at detector 0 would not be possible)
Detectors 1 and 2 get two light beams each. One beam reflected twice which means the light incident on each detector is a combination of left and right hand circular polarised light beams. Which can only result in plane polarised light hitting each detector 1&2. (Note that if one combines a left handed circular polarised beam with a right handed circular polarised beam the combined result is a beam that alternates between two opposing polarised states and plane polarisation each cycle. That is effectively plane polarised light.
Because the D1 beams undergo opposite reflections at the mirrors and beam splitters from the D2 beams...the two detectors 1&2 thus also each receive polarised light that is phase shifted by half cycle from the other. Ie..vertical and horizontal respectively.
Detectors 3 and 4 only get one beam reflected once, each.(The one reflection restores the polarity of the beams from orthogonally polarised circular light leaving the BBO/G-T prism setup back to identically circular polarisation. Which means they get the same phase light as that arriving at detector zero.
The resulting polarised states mean D1&2 each only “observe at Detector Zero” via the coincidence counter two seperate phase shifted interference patterns from the other .And detectors 3 & 4, having the same incident polarised light as detector zero , observe diffraction patterns via the coincidence counter.
No spooky quantum mumbo jumbo maths needed to explain this experiment.
And finally to address claims that the Glan Thompson prism used in the experiment cannot send circular polarised light beams to the rest of the experimental setup. Critics cite evidence showing that similar birefringement mediums such as calcite crystals split the beam into two orthogonally plane polarised beams. And using plane polarisation filters they show how the two exiting light beams must be only plane polarised. As a plane polarised filter put over the two images only allows through one of the two images coming through the calcite filter.
However this ignores the fact that a vertically plane polarising filter allows through not just vertical polarised light...but all angles except horizontal. So for instance around 45% of the light polarised at 45 degrees to vertical is let through a vertically polarised filter. The more the polarisation angle of the light deviates from vertical the less light is let through etc.
In other words a circular polarised beam will be split into two polarised beams by the crystal one beam preserving majority vertical, one majority horizontal. But still containing elements of all other angles of polarisation. And thus the circular polarisation of the beam is preserved.
Proof of this is available at any demonstration showing unpolarised light going through two plane polarised filters. Position both filters at vertical and all vertically polarised light from a source goes through, and blocking all horizontal polarised light. Turn one filter slowly to horizontal and the light coming through decreases to zero as both vertical and horizontal light from the source are now blocked. Light coming through doesn’t immediately decrease to zero as soon as the two polarised filters angle starts to diverge. It is an incremental decrease. Proving a plane or linear polarising filter still lets through almost all angles of polarisation. But at different amplitudes.
This is how one can explain how two circular polarised light beams can appear as two plane polarised beams exiting the Glan Thompson prism.
For further description of how this can be modelled as a classical effect only, watch the following...
https://www.youtube.com/watch?v=_KekfbrzO74