Your original “pick two” comment implies that you can pick “anonymous” + “free of bots”, or “available to everybody” + “free of bots”. I can’t think of any system that would satisfy those.

The former can be had on a system accessible only by a restricted group of people, through terminals that have no automation interfaces. Think of a local private library with a bulletin board: Only members can enter, and once inside, anyone can post using the provided tools, but there are no tools that bots could use, and there's nothing connecting an individual to what they post on the board. Of course, a system like this would not be available to everyone; it wouldn't work as an internet service, for example.

The latter can be had by requiring an access token issued to each user. Bots could then be either ejected quickly by revoking their owner's token when discovered, or prevented early by requiring all access to be through public terminals that have no automation interfaces. Of course, any system that issues an access token to an individual cannot be anonymous if revocation is to be meaningful.

The government would keep track of whether or not they issued you an ID,
The token/ID system adds anonymity

I wish that were possible, but no, it is not. If a government (or any other authority) keeps track of whether they issue someone an ID, then they must also have the ability to connect IDs to individuals. Such an arrangement can only be anonymous if everyone involved is a good-faith actor choosing not to abuse that ability, which history demonstrates is not a realistic assumption.