>>> Thanks, this fixed the issue.  And I noticed another one: at the
>>> beginning of the buffer previous-completion wraps to the end of the buffer,
>>> not to the beginning of the last completion.
>>>
>>> Also when point at the last completion, next-completion
>>> jumps to the end of the buffer instead of wrapping to the
>>> first completion.
>>
>> I cannot replicate this issue in every case, but only if the last option
>> has completion annotation.  It should be fixable by checking for these
>> kinds of edge-cases, but it might also be better to reconsider if
>> next-completion should be using `mouse-face' for navigation, or if a
>> special text property might be more elegant?
>
> I agree that `mouse-face' is unsuitable for detecting the completion
> boundaries, and it causes other problems: typing RET on the completion prefix
> or on the completion suffix/annotation causes the error:
>
>   Debugger entered--Lisp error: (error "No completion here")
>     error("No completion here")
>     choose-completion(13 nil)
>     funcall-interactively(choose-completion 13 nil)
>     command-execute(choose-completion)
>
> So a special text property could also help to detect the completion for RET,
> if there are no such text properties already.  But I see there is the
> text property 'completion--string'.  So maybe it could be extended to cover
> prefix and suffix/annotation as well.

Here is the patch that fixes 4 bugs: 2 bugs described above,
and also bug#55289 and bug#55430.