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I need to record video projected onto a wall or screen by an Optoma EP726 digital projector. I need to record it on a Canon EOS 77D at 23.976fps.

I first tested videos of various frame rates, all exported and projected at their native frame rates -- I took a single video, modified its frame rate in Premiere and made sure each sequence matched the video's new frame rate -- and did all the recording at 23.976fps, the frame rate I'll need to use for the real project.

Here are the results of those tests, all 10-20 seconds long, all shot with a shutter speed of 1/100s:

Video projected at 23.976fps, recorded at 23.976fps, 1/100s:

Video projected at 24fps, recorded at 23.976fps, 1/100s:

Video projected at 25fps, recorded at 23.976fps, 1/100s:

Video projected at 60fps, recorded at 23.976fps, 1/100s:

All the videos are black and white, but when I record the projector's output, you can see moving bands of red, green and blue showing up in the camera's footage. Those are visible on the camera's preview screen, but not to the naked eye. The bands remain visible even when the resulting video -- that is, the video out of the camera -- is made black and white.

The following setups were more successful, and I think two of them have no banding at all.

This test was projected at 25fps and recorded at 23.976fps, but with a shutter speed of 1/25s, and the problem is still visible but it's only slight:

This test was projected at 23.976fps and recorded at 23.976fps with a shutter speed of 1/25s, and as far as I can see there is no problem, but please do correct me if I'm wrong:

Finally, this test was projected at 23.976fps and recorded at 23.976fps with a shutter speed of 1/60s. I would have expected bad banding given the high shutter speed, but in fact I don't see any:

This was baffling me at first and I started this question intending to ask how to fix it, but apparently it can be fixed by making sure the projected frame rate, camera frame rate and camera shutter speed align so that the camera preview shows no moving bands. I'm posting this as a question anyway because it might be useful to others.

There is still a less important question, though: I don't know how to determine a priori what combinations will succeed. Practically speaking, I'll just use one of the two successful test setups. But if it could be explained how the interference pattern works, I'd like to know. I imagine it's possibly also connected to the AC frequency. I'm in the UK, so I believe it's 50Hz.

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The banding occurs because the projector uses DLP technology - it rapidly switches between red green and blue output faster than most people can notice.

Your camera has a rolling shutter, and sometimes the different colour bands are caught by the camera if the shutter and projector aren’t perfectly in sync. Ideally you would sync the projector and camera with the same clock, and/or use a projector that uses LCD instead of DLP.

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You are correct that is is connected to the AC frequency, the refresh rate on projectors was almost always the same rate. HOWEVER, most projectors (most electronics) at this point have switch mode power supplies, which means the devices are no longer locked to the mains frequency. So most devices at this point operate internally at 60Hz, not sure if that's due to being manufactured in 60Hz countries or some other reason, but that's mostly what you'll find. Which is why your 1/60 shutter speed sync'd up. If you're getting a rolling effect at 1/60, then 1/50 should be the one that works.

1/25 probably looked OK because you were at less than half the refresh rate of the projector, no strobing, but you're catching slightly over 2 exposures from the projector, which may cause other issues.

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