About the only thing I don't refrigerate after opening is peanut butter. You'd never be able to get it out of the jar if you refrigerated that.
It stays in our refrigerator. Chic fil a keeps the packets on a counter, unchilled, albeit unopened…For 49 years wife and I have had a difference of opinion. Do you refrigerate ketchup after opening or not. I say leave it out something that acidic won't spoil, she say refrigerate like Mrs Kerry said before we dumped her.
BTW The ketchup is in the fridge.