We should rename moral realism and antirealism
Moral "realism" and "antirealism" are unhelpful terms. They don't clearly distinguish the positions different people take in metaethics, and can give people misleading impressions of what opposing sides dispute.
Unfortunately part of the problem is that one can reject non-naturalist moral realism for different reasons than one rejects naturalist realism. This makes moral "antirealism" a somewhat odd position: It's defined in terms of rejecting (at least) two distinct realist views which themselves often reject one another on similarly strong grounds. Furthermore, people sometimes presume that realists think morality is "real" and antirealists think it's "not real," leading to some pretty strange assumptions about what antirealists think. An antirealist does not necessarily think "morality" isn't real; they don't, for instance, necessarily deny the existence of moral systems, moral beliefs, moral attitudes, moral judgments, moral reasoning, and so on (though they might deny some of these). Rather, they tend to deny both naturalist and non-naturalist accounts of moral realism.
Naturalism could also mislead people: it's not as if, as an antirealist, I don't think morality is a natural phenomenon. But like non-naturalist realists, I don’t think naturalist programs are going to work, because I think they can’t capture essential elements of what’s philosophically in dispute. In particular, they don’t strike me as capable of capturing the normative element of moral claims, and as a result end up providing us with descriptive claims which we may simply not find motivating, have no authority over us, etc. Perhaps we should rename or reframe the terms used in these disputes to avoid misunderstandings and make different accounts more accessible to those who don’t study metaethics. Perhaps we could have four terms, which could capture endorsement and rejection of non-naturalism, and endorsement and rejection of naturalism about moral facts.
Perhaps would yield the following positions:
Metaethical Mooreanism
Metaethical Anti-Mooreanism
Metaethical descriptivism
Metaethical anti-descriptivism
These may not be the best terms, but they may be better than the current set of three positions: non-naturalist realism, naturalist realism, and antirealism.