Fake bans aren’t a thing, but banning from an instance doesn’t necessarily ban them from all communities (just the ones they’ve interact with up to the time of the ban).

This means someone on instance B, who is banned from instance A, can still interact with communities on instance A via instance B if instance B was not informed the person was banned from the community.

However, a user from instance B, banned from instance A, interacting with a community on instance A via instance B will not have their content federated via instance A (the home instance of the community). Thus, only other users on instance B should be able to see that user’s submissions.

Clear as mud?

Edit: Maybe I can clear it up with an example.

Let’s say you and your friend are both on lemmy.cafe. I see your friend around in the Lemmyverse from my instance, and they’re violating a bunch of my local rules to the point they merit a ban. Your friend has not interacted directly with any communities on my instance, so when I ban them, no community bans are federated.

You and your friend are both subscribed to, say, [email protected]. Even though your friend is banned from my instance, they can still post/comment there from lemmy.cafe since that instance has no idea they’re banned from mine; non-local bans aren’t federated, and since they hadn’t interacted with that community prior to me banning them, no community bans were sent out.

However, none of the posts/comments they make to lemmy.cafe’s local copy of that community will be federated by my instance (where the 30 Rock community lives). Thus, only users on lemmy.cafe can see what the banned user has posted.