I think this cropped up when I changed my secret key. My laptop was recently stolen and in the aftermath I changed my secret key. Everything seems to work fine on my new laptop. The Mac app, the web app, and the 1Password X extension all behave normally. My desktop machine, different story.
The desktop was setup before the secret key change, and 1Password X is acting strange. The Mac works fine. 1Password X works correctly but it won't sync, so it has a stale view of my vaults. When I login to 1 Password X, it works, but I immediately get a Chrome popup telling me "Authentication Required". If I do what it says, follow the link, and login to the web app, this works fine, but the X extension still does not sync, and the next time I login to the extension the same thing happens. Following advice I saw elsewhere I've tried logging out and back in again to the web app, but nothing changes.
This is with Chrome on the Mac. I also recently set up 2FA. Not sure if this started when I change the secret key, or when I turned on 2FA. Don't remember the exact order. Something else weird is that when I go out and back into the web app, I am not getting prompted for 2FA.
1Password Version: 7.2.5
Extension Version: 1.14.2
OS Version: 10.13.6
Sync Type: Not Provided