TimeMachine has stopped backing up due to "Anti-virus software may be preventing Time Machine from writing to the backup"

TL;DR - Problem Solved!

Details (probably more than you bargained for!):

• To recap, I'm running QNAP NAS devices as backup servers at both of our locations. TimeMachine backups to the one here stopped working on my Mac (Monterey), while still working on my wife's Mac (Big Sur). I would receive an error message implicating "anti-virus software", which was (and still is) confusing.
• Running TimeMachine to a local USB disk on my Mac worked fine, which pointed me back to the NAS.
• This portion is pretty QNAP specific, but does have some TimeMachine-relevant nuance:
  • Examining the files on the QNAP, I noted that my wife's TimeMachine "sparsebundle" folder was visible both in File Manager and in Hybrid Backup Sync (HBS) (a QNAP-specific backup-server-management app). However, my "backupbundle" folder was visible only in File Manager, not in HBS. Strange.
  • I did not go down the "sparsebundle" v "backupbundle" rabbit hole.
  • Given that this NAS backup was not critical for me, I went ahead and deleted the "backupbundle" folder, dismounted and re-mounted the TimeMachine volume ("Connect to Server..."), re-connected to the NAS TimeMachine volume within TimeMachine, and allowed TimeMachine to create a new backup.
  • This is working as expected! TimeMachine is now creating the initial backup.

So, problem solved. It's still a bit of a mystery to me exactly what happened, whether it was simple file corruption, some issue with QNAP's handling of "sparsebundles" v "backupbundles", or some issue with Apple's handling of legacy backups, or .... 

@steve626 - thank you for that! As I nose around on the machine, I don't see Avira having been installed (though that .kext file is definitely listed in the EtreCheck output; I note that it's tagged as "[not loaded]". There is no Avira application or folder in the Applications or Utilities folders, so I suspect never installed? (The files date from 2015; must have come across during migration to the new Apple Silicon machine.) Avira's uninstall instructions (https://support.avira.com/hc/en-us/articles/360000759429-How-do-I-uninstall-Avira-Security-for-Mac-) presume that there is an app installed. I did see this article in the discussions forum (How do I completely remove Avira? - Apple Community), which I may have to resort to (though I'm concerned that some of the manual operations may be above my pay grade).

As a first attempt, I have downloaded the Avira installer, installed the app, and uninstalled it. Rebooting and re-ran EtreCheck to see if that has cleaned up the zombie .kext file. It is no longer listed under “Kernel Extensions”, so I trust that it’s gone. There were a few other breadcrumbs (per the discussions post above about completely removing Avira) - deleted them. Allowed TimeMachine to attempt a backup - same error as above. Dang!

Here is latest EtreCheck output, post Avira install/uninstall routine

(attached).

Chapeau to @Steve626 - TimeMachine working Just Fine to my shiny new USB disk drive (directly connected to my Mac). Failure mode persists on the NAS-based archive, despite (again) re-booting NAS and dis-mounting / re-mounting the TM volume's access from the Mac. NAS TM backups continue to work fine from my wife's Mac (she is on Big Sur, not Monterey).

Now working with QNAP community and support to figure out whether and, if so, how the NAS is failing to interact properly with TimeMachine. Stay tuned!

@OldToad - thanks for that, and for the pointer to Find Any File. What a lovely little utility! (Unfortunate acronym, though.)

Ran that and it did, indeed, find many more breadcrumbs, including a hidden "Avira" folder in Applications. Deleted all of that; emptied trash. Re-ran FAF - no further instances of "Avira" on the machine.

Re-booted: same problem.

Re-booted in Safe Mode: same problem.

I am stumped.

Okay, today's update:

• Updated firmware on the QNAP NAS. No change in behavior on my machine. Recall, though, that my wife's machine (running Big Sur) backs up to that same NAS using TimeMachine with no difficulties. Also, my machine was backing up to that NAS just fine until 6 September. Checking the logs on the NAS I don't see any salient events on/about 5 or 6 September (just me logging in as Admin).
• Updated Mac to 12.6 (from 12.15.1). No change in behavior on my machine.

@Steve626 has suggested running a backup to a hard-wired disk. I neither own nor have access to one, but at this point (2 weeks without a reliable backup), it's feeling prudent to get one if only for use until I/we figure this out. So, I'll pick up a disk (will take a day or two), try that, and report back.

[Side note: I note that a couple of other folks appear to have run into this TimeMachine error (based on posts to the Discussions fora) - I don't see any of those marked as "solved", so maybe this is some weird corner case?]

[Side note: Lastly, and in the realm of pure speculation, I suspect that the Avira stuff showed up on the scene as a "gift" from DropBox. I note (based on Wikipedia), that Avira and DropBox announced a partnership at about the same time that Avira appeared on my machine (2015/2016). No doubt I allowed this gift, but don't recall it specifically. Not germane to the troubleshooting, but thought I'd share it.]

I'm sorry - please ignore previous EtreCheck report. This time with Full Disk Access enabled (doh!).

Hopeful that someone will have some insight here. Next stop: Apple Support, I guess....

Thanks for your note. As I've recounted above, I did remove Avira and its breadcrumbs, assisted by Old Toad and Steve626. Problem persists. Trying some other things this morning and will report back.