Earlier quoted context omitted.
Mm, I think it could come up quite naturally as "she must have been being watched" in something like: Two detectives are watching security film. DETECTIVE 1 And then from 9:07 to 9:14 she started acting very cautiously, but – Detective 1 rewinds the tape by a few seconds and gestures at the screen. DETECTIVE 1 – from this angle you can tell that she's making an effort to not seem suspicious. I wonder why? DETECTIVE 2…
I have read the “must have had been being” over and over again and as a native English speaker, I still can’t understand what it means. I hesitate to call it ungrammatical, but instead throw down a challenge: can you actually use it?
She must've been being watched.
suggests that, at that point in the security footage, she might still be being watched. If the detective had said She must have had been being watched.
it would additionally imply that, at that point in the security footage, she was no longer being watched (but had been). So let me rework the story a bit: Two detectives are watching security film.
DETECTIVE 1
She's acting ordinarily, but I see fatigue written on her face. I wonder why.
Detective 1 leans back in their chair and muses.
DETECTIVE 2
She must've had been being watched. The state security service monitors high-profile civilians occasionally, and a few days before this she was acting very cautiously, almost as if she was suspicious of something.
(Also, I think I falsely portrayed the universality of such a construction. Although I think some native English speakers would accept it as grammatical, many might not!)