For example, let's say that you are a typical Ashkenazi who will have some Gefilte Fish as an appetizer on Shabbat. Obviously, we won't have meat at the same time. Some people will rinse their mouths out afterwards, before eating meat, and still others will wash their hands between courses. And, of course, any silverware that even touched the fish will have to be removed, even if you cannot see any evidence of fish upon it.
In other words, we expanded the tradition tradition of not eating fish and meat together to be as though a single drop of fish essence upon the meat will cause your skin to flake and rot, even though the sages in Tractate Pesachim stated that if one cooked meat in a pot that already was used to cook fish, then that's fine. The problem they had was in cooking them together (because of that original health diagnosis). So the reality is not that we have to be neurotic about it, but that you don't eat fish and meat together as a mixture. And who knows, maybe the original statement in Psachim was metaphorical, since they also taught that tz'raat is a spiritual, rather than a physical affliction.Given a chance to be neurotic, we leap at the opportunity!
As a final note, some people have added another custom to drink some liquor between the fish and the meat meal to "rinse out the palatte", as though anyone really needs a reason to do a L'chaim! And perhaps this was added because of the error in thinking that a fish and meat mixture caused tza'raat.
So, in reality, I do not eat meat and fish together, and I even support the tradition to take away the plate and fork that was used on the fish before bringing out the meat, but excuse me if I don't shriek out a warning if someone is using the same knife on a piece of turkey that was used on the fish. It's just weird to treat it as if tz'raat will immediately spread if you don't act with zeal.
