Eliezer is a realist, he's just also an indexicalist. According to his theory, when you use the word "morality", you refer to "Human!morality", and there are objective facts about that. His theory just also says that when Clippy uses the word "morality", it refery to "Clippy!morality" (about which there are also objective facts, which are logically independent of the facts about "Human!morality"). Just like when you say "water", it refers to water, but when twin-you says water, it refers to XYZ.
Group-level moral relativism just is the belief that moral truths are indexed to groups. Since relativism is uncontroversially opposed to realism, "indexical realist" is a bit of a contradiction.
Continuing my quest to untangle people's confusions about Eliezer's metaethics... I've started to wonder if maybe some people have the intuition that the orthogonality thesis is at odds with moral realism.
I personally have a very hard time seeing why anyone would think that, perhaps in part because of my experience in philosophy of religion. Theistic apologists would love to be able to say, "moral realism, therefore a sufficiently intelligent being would also be good." It would help patch some obvious holes in their arguments and help them respond to things like Stephen Law's Evil God Challenge. But they mostly don't even try to argue that, for whatever reason.
You did see philosophers claiming things like that back in the bad old days before Kant, which raises the question of what's changed. I suspect the reason is fairly mundane, though: before Kant (roughly), it was not only dangerous to be an atheist, it was dangerous to question that the existence of God could be proven through reason (because it would get you suspected of being an atheist). It was even dangerous to advocated philosophical views that might possibly undermine the standard arguments for the existence of God. That guaranteed that philosophers could used whatever half-baked premises they wanted in constructing arguments for the existence of God, and have little fear of being contradicted.
Besides, even if you think an all-knowing would also necessarily be perfectly good, it still seems perfectly possible to have an otherwise all-knowing being with a horrible blind spot regarding morality.
On the other hand, in the comments of a post on the orthogonality thesis, Stuart Armstrong mentions that:
This is not super-enlightening, partly because Stuart is talking about people whose views he admits he doesn't understand... but on the other hand, maybe Stuart agrees that there is some kind of conflict there, since he seems to imply that he himself rejects moral realism.
I realize I'm struggling a bit to guess what people could be thinking here, but I suspect some people are thinking it, so... anyone?