I checked again with FireFox, TorBrowser, DuckDuckGo, LibreWolf; this time with all their cookie restrictions turned off completely, I still get the same result.
If you go directly to the Archive, you won't get past the age verification.
If you go to the Forum, don't click anything, then go directly to the Archive, you won't get past the age verification.
If you go to the Forum, use a link to "https://adult-fanfiction.org/", from within the forum, then the age verification works.
I've tried this on Android 13, Windows 10, Zorin 18 and Debian 13; A phone, two PCs and a laptop.
I think there may be something wrong with the Age Verification prompt, at least in regards to browsers based on FireFox, because no other site acts that way for any of my other devices or browsers.
At the very least, maybe there should be a warning on the age verification that it may not work with browsers based on FireFox.
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Am I the only FireFox user here?