I’ve done some further investigation, and it seems like it’s actually a NuGet issue that’s unrelated to the (Monogame) NuGet feed; I’ve got several NuGet feeds configured, and one of them requires authentication.
However, for some reason, cancelling out of that, (since I don’t use that feed for the monogame project) actually interferes with NuGet’s query to the teamcity server, causing NuGet to request authorization for the MonoGame feed, where anonymous access is actually allowed.
Apparently, me cancelling out of the authorization for one feed disrupts the authentication flow for another.