My intent was just to highlight that realism, non-naturalism, universalism, and primitivism are different ideas. I wasn’t weighing in on their relationship, beyond their non-identity. Universalism and primitivism, for instance, I’d usually consider compatible with an error theory of morality (and thus with anti-realism): Moral statements are semantically irreducible or structurally applicable to everyone, but fail to meet their truth-conditions. Similarly, I could imagine people committed to anti-realism precisely because moral facts would have to be non-natural. We may not want to call the latter view ‘moral non-naturalism,’ though.
My intent was just to highlight that realism, non-naturalism, universalism, and primitivism are different ideas. I wasn’t weighing in on their relationship, beyond their non-identity. Universalism and primitivism, for instance, I’d usually consider compatible with an error theory of morality (and thus with anti-realism): Moral statements are semantically irreducible or structurally applicable to everyone, but fail to meet their truth-conditions. Similarly, I could imagine people committed to anti-realism precisely because moral facts would have to be non-natural. We may not want to call the latter view ‘moral non-naturalism,’ though.