About 313,000 results on a Google search for "mustn't he" would perhaps indicate that people saying the usage is terrible (1) haven't checked and (2) are speaking subjectively.
UsingEnglish.com has an article addressing this debate:
He must have seen it, mustn't he?
This is the first time I have seen such a question tag as with 'must
have done'. In China's English tests or examinations, such a tag is
thought to be absolutely wrong. They say we must use 'hasn't he?'.
They also say we must say 'He must have seen it yesterday, didn't he?'
I thought 'mustn't he?' was fine, but I had never found any proof. Now
that I've got this sentence, I'd like to know how authoritative it is
and whether we can use this sentence at all.
Might I ask native English teachers to help me clarify this puzzle of
And after checking, as well as being familiar with the usage, I can agree with the reply [bolding mine]:
There's nothing wrong with this. Any other tag question implies an ellipsis such as 'He must have done it. [Tell me I'm not wrong.
(Beginning to be unsure) He did...] Didn't he?'. In that case, it's
not really a tag question at all, as it's not asking for routine
confirmation of a certainty.
The modal tag-question is very common:
I can go out tonight, can't I?
They could go via Vail, couldn't they?
It would be a disaster, wouldn't it?
He must use the lift, mustn't he?
I shall die, shan't I? [formal]
He'll tell his dad, won't he?
'Mightn't' may sound a little awkward, but 'mustn't' is in regular use, at least in some areas. ODO includes the contraction and a tag-question usage:
So I thought, if it's man made, there must be someone driving it,