[agalunar]

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
  She must've been being watched.

I do agree that form "must have had been being —ed" (with "had" in there as well) is an especially rare form, but in context, native English speakers understand what it means!

I wonder how many other constructions there are that native speakers don't really produce but that they still consider grammatical or as not even being unusual.

[ickoonite]

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?

[agalunar]

Sure! Using "had been" rather than "(has) been" indicates that something may no longer be the case, e.g. "she's been well (and still is)" versus "she had been well (but is not necessarily still well)". So in my example

  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!)

[ickoonite]

No, this is just wrong. You are conflating two tenses. Must can be present (be) or perfect (have).

In the example above, you would say: “She must have been being watched.”

[agalunar]

To my ears, there's a difference!

  I am watched.            (present passive)
  I was watched.           (past passive)
  I have been watched.     (present perfect passive)
  I had been watched.      (past perfect passive)

  I am being watched.      (present progressive passive)
  I was being watched.     (past progressive passive)
  I've been being watched. (present perfect progressive passive)
  I'd been being watched.  (past perfect progressive passive)

  I must be watched.
  I must have been watched.
  I must have been watched.
  I must have had been watched.

  I must be being watched.
  I must've been being watched.
  I must've been being watched.
  I must've had been being watched.

I think is roughly it? Although now I'm beginning to doubt myself, and maybe it is just:

  I must be watched.
  I must have been watched.
  I must have been watched.
  I must have been watched.

  I must be being watched.
  I must've been being watched.
  I must've been being watched.
  I must've been being watched.

or really, maybe it's better to say those forms don't exist:

  I must be watched.
  I must have been watched.
  —
  —

  I must be being watched.
  I must've been being watched.
  —
  —

What's neat is that I can assign meaning to "must've 'd been", but it really teeters on the edge, sometimes sounding strange but acceptable and sometimes sounding simply wrong.

I can't find many examples of it, but there are a few I've found online:

"To qualify for a special enrollment due to a permanent move, you must have had been enrolled in other minimum essential coverage, such as under a job-based health plan, another Marketplace plan, or Medicaid."

"Sales must have had been in the same year as the tax return."

So perhaps it's best to say it's nonstandard but attested.

[ickoonite]

Oh my. You have got yourself in quite a pickle. “Must have had been” is simply wrong; in your examples, it should always be “must have been”.

[agalunar]

> "Must have had been" is simply wrong; in your examples, it should always be "must have been".

As a matter of prescription, maybe, but as a matter of description? I think it's interesting. It is meaningful and it's something that English speakers produce. Here are other examples in the wild:

"To complete the request, the owner of the permit must have had been present at the time the citation was written, present a valid permit to the Tax Collector in person, have a copy of their citation, and pay a $7.50 processing fee to the Tax Collector if approved."

"Lankeshwar must have had been easy to defeat compared to new-age Ravanas."

It can even have a distinct meaning (rather than just being a variation), which is how I would interpret it. Which I think is cool.

It's perfectly fine to say "don't use this construction if you want to be taken seriously by so-and-so", but if its usage happens to be dialectal, for example, I don't think you can say those speakers are using their dialect of English incorrectly.