A similar explanation, the "ideomotor effect," has been put forward for the Ouija board and several other divination techniques that sometimes seem to do better than chance or produce other kinds of "spooky" results.
The idea is that you have subconscious thought processes, sensory awareness, or knowledge, and when you put yourself into an observation/motor feedback loop this subtile cognition can influence fine motor responses which are then amplified by feedback as you observe your own movements. You're putting weak neural "noise" through a feedback amplifier, but the noise is not random and could contain information. There is no way to control for this because the test is not blind.
It's become popular in recent years to chase "chains of synchronicity." A very entertaining semi-documentary about this can be found here:
https://www.hellier.tv
There's even an app for it called Randonautica.
A very similar phenomenon could be at work in that. It might start with subconsciously biased paradolia and as you follow a "chain of synchronicities" you are running an amplification and filtration feedback loop similar to what happens when you watch a planchette move across a Ouija board or watch a dowsing rod move. You've grabbed a set of weak neural signals and amplified them.
It's not supernatural but it might be neurologically and cognitively interesting. I've had the intuition for a long time that all creativity probably involves processes like this. An artist or a musician doesn't just emerge with a fully formed work from nowhere. They "noodle around..."
BTW...
Hellier is a great bit of independent film. I've compared it to the cult sci-fi film "Primer" in terms of quality, originality, and creativity on a shoestring budget, and if you liked that film you'll probably like it. Also heard it described by someone as "like Ghost Hunters or Paranormal State but for thinking people." It's a fun ride if you suspend disbelief. There's a scene in season two where a practicing witch uses a transcranial magnetic stimulation device to try to talk to aliens. It's worth it just for that, gives it a hint of "post-cyberpunk."
Edit:
Has anyone ever experimented with these techniques as a cognitive or creativity enhancer without the "woo woo" assumptions? Treat it like a brain hack to get at subconscious thought processes, sort of like what you can do with psychedelics but without the drugs.