Since realism and relativism are typically (and erroneously) considered opposed and incommensurable metaethical positions, a little brush-clearing is in order before I attend to considering whether Nietzsche is both. So:
Moral realism is the metaethical thesis that moral claims are either true or false.
Moral relativism is the metaethical thesis that the truth or justification of moral claims is relative to some group or individual. By convention, the truth or justification in question is taken to turn on presumably arbitrary (if in some sense "natural") cultural norms or personal convictions.
Two points. First, even on the conventional account of relativist justification, a moral relativist could be realist about moral claims. You'd simply need an argument to the effect that a proposition like "It is wrong for Bob to have nonmarital sex" is true (if it is true) in virtue of certain objective facts, viz., either that Bob is a member of a well-functioning moral community in which the prevailing mores prohibit nonmarital sex, or that Bob harbors personal, suitably inhibiting convictions about nonmarital sex.
Second, and more importantly, there is little reason to suppose that moral claims must be true or justified relative to arbitrary cultural criteria as in the standard account. There are lots of features of the world to which moral claims could be relative. Time. Language. Geography. Season. Planet. Gym membership. But since the fundaments of moral choice and action reside in human psychology, it would at least seem more plausible than any of these to suppose that moral claims are relative to the psychology of moral agents -- their motivations, dispositions, potentials. Call this "psychology-relativism," or "p-relativism" for short.
Now, it's quite clear that if Nietzsche stakes out a realist-relativist position, it is not along the lines of standard relativism. Nietzsche takes himself to be standing quite outside, or above, the culture into which he happened to have been born; prevailing cultural norms, then, so far from being a source of normative guidance to Nietzsche, are objects of his derision and devaluation. Similarly, Nietzsche thinks little of convictions; they are merely what results when "mental sloth lets [opinions] ossify." (HAH §637) So standard relativism is out.
P-relativism, on the other hand, squares pretty well with Nietzsche's notion that there are "higher men" toward whom only a distinctive, aristocratic sort of "thou shalt" is proper. (Cf. D 4) His advocacy on this score evinces a clear commitment to the existence of something objective in such persons, something that renders them noble, admirable, laudable, exemplary of human greatness. And crucially, he wants to say that such greatness should in some way be normatively gripping (at least for those equipped with the "right ears"). If Nietzsche is a realist-relativist, then, he is some kind of p-relativist.
