Nietzsche clearly evaluates past moralities as constructive or destructive, and expresses strong feelings about the goal of a life from 'One must make of one's Self a work of art' to 'Man is what must be overcome' and boundaries on action 'Fighting monsters we must not be made monsters'.
He expresses these as truths. And he feels urgent value from the eternal recurrence he deduces from Newtonian physics (if incorrectly, with a classical sense of math.) It is hard to see him, then as a total skeptic relative to shared facts, even facts about values. Even if, in the end he prefers an aesthetic sense of truth over some obligatory objectivity, he feels aesthetics have a certain kind of objectivity, or he would not bother to quarrel with Wagner.
A lot of 'critical theory' people launch from there and simultaneously undermine or devalue all of our shared standards of aesthetics as well, and that leads to the notion of global anti-realism. But they cannot cast that whole debt back on Nietzsche, they generally need to pull in Marxism, psychoanalysis or other leverage to accomplish the more important half of the task.
From what I understand of Leo Strauss position on Nietzsche, he thought that Nietzsche had to be an outright nihilist by assuming that he did not accept myth as a form of truth. This is much the same thing I see critical theorists doing, but he is using the remnants of an abstract Platonism that he himself rejects, to undermine the sense of realism in myth.
It is true that Nietzsche explicitly describes these as contrasting categories, but clarity of language often suffers from art. He can't mean this too wholeheartedly, because he then fights very hard on the behalf of his own myths. To my mind, relativism is not nihilism if one accepts that shared myths and aesthetic standards will speak to others in a meaningful way, and that they are worth maintaining for that purpose. 'Choose your delusions' is a form of relativism as an exercise in political aesthetics that is therefore not nihilistic. It can, in fact lead to very strong religious sentiments, in a way that real Empiricism or Pyrrhonism cannot.