This is not a question about academia per se, but about the academia-administration interface.
When I was a postdoc, I wrote a paper that started to be cited quite often (by the standards of my particular subfield) as soon as I uploaded it to the usual public repository. These were largely good citations, along the lines of "this is a really neat idea" rather than "this author doesn't know what he's talking about". Eventually, I decided against submitting it to a journal, mostly because the unpublished manuscript had already become relatively popular and it wasn't really worth going through the extra work of the journal publication process for essentially no additional benefits re: the visibility of the paper or my reputation in the field. I've been listing it in my CV as a "frequently-cited unpublished manuscript" and everybody has been happy about it.
Now that decision is coming back to bite me in the ass. I am applying to a tenured job and I have to submit copies of representative publications. However, I can't submit a copy of this paper specifically because it has never been published in a journal. I have enough quality publications that this is not a problem, but it annoys me because I'm quite proud of this particular paper, which is as representative of the quality of my work as any of my papers in top journals.
So the question is, those of you who have similarly popular unpublished manuscripts, how do you convince administrators that a paper doesn't have to have a journal DOI attached to it to count as a representative piece of work?