I run Microsoft Office on my iMac for files on which I need to collaborate with others (inexplicably) not using Macs.
Yesterday, copies of two old Excel spreadsheets (.xlsx) and nine Word documents (.docx) appeared unexpectedly on my iMac’s desktop. Each file name was prefixed by a tilde ~, and suffixed by “ineligible”. I had not made these changes, and had not opened these files for a long time. They were saved in Dropbox rather than OneDrive. Unaltered originals of these files remain in their original Dropbox folder, so I have deleted all but one of these mystery copies from my desktop. One of the adulterated files refuses to be moved, renamed or deleted.
This strange “fault” has not caused me problems, but I do not like things like this happening for no reason. Checks suggest I have no malware.
Can anyone suggest what may have happened?
It might be an Office update that could have put them there because they might be incompatible with the new version?
Those are temporary files Microsoft Office products create and use as long as Word documents or Excel files are being in use. Those files are being closed and deleted automatically when Word or Excel files are closed in a proper manner.
More on the background of temporary files being created by MS Office:
If those files are not deleted automatically, something went wrong while closing a file.
You mentioned “collaboration” and you mentioned “Dropbox” and “OneDrive”. Those are three key points that can be a reason for temporary files not being deleted and remaining “in use” (syncing conflicts, permission issues and what not). I have no idea why you ended up with files from your Dropbox on your Desktop. That is something to debug depending on what you did 
and what your settings are (in your “cloud” apps like Dropbox or OneDrive). 
Feel free to delete those “tilde office files” after making sure that you have closed all Office apps.
If you are not able to delete those files, there is some kind of file corruption going on that could be blamed on the cloud solution of your choice or the operating system as a result of issues mentioned previously.
Try deleting those files using Terminal (handle with care):
Or with even more permissions via the Terminal (handle with care):
Or try deleting using OmniDiskSweeper:
You mentioned collaboration: Collaboration can mean that several users work on the same file simultaneously. This is one of the main causes for temporary files that are not being deleted properly. Regarding Dropbox: if Dropbox syncs also those temporary files, you very well may end up with temporary files from other users…
Microsoft expects you to collaborate (= simultaneously on the same file) using their cloud solutions these days:
Dropbox is fine as long as you make sure that Word and Excel files are not accessed by several users or on several computers simultaneously.
Thank you for a very helpful answer. I have some research to do during the weekend,
I should have added I have used the same file storage configuration for years and have never seen this before.
I use iCloud for certain files and certain apps on which I am working actively. Once they are complete I move them to Dropbox.
I keep some active files only in Dropbox, including some on which I collaborate.
I have OneDrive as part of my Office 365 subscription, but rarely use it. Perhaps I should use it rather than Dropbox for active Office files on which I am collaborating.
Anyway, I have not lost any files and this was just an occurrence that I had never seen before.
Assuming that your Desktop and your Documents do live on iCloud (Add your Desktop and Documents files to iCloud Drive - Apple Support):
If those files were located on your (iCloud) Desktop and you opened them there and if there was an issue later on (with iCloud or while working with those files) all of this does make sense. Maybe even iCloud file sync revived those files for some reason. Nothing bad. And for sure no malware.
Don’t change anything. I do not see any issues in your workflow. My guess is that something acted up temporarily while those files were stored on your Desktop and in use by yourself in Word / Excel. Maybe iCloud-related, maybe OS-related, maybe Office-related. Nothing to worry. Something happened that caused those temp files not to be deleted or to resurface. 
Do not waste too many thoughts on this, delete the files and you should be fine.