What leads the moral realist to assert that mind-independent moral values actually exist?

In your original post you say

> It follows from the actual attitudes of realists that there are independent normative standards.

Later you say

> the non-naturalist realist will happily agree that ... [belief in mind-independent moral values] lacks grounding in empirical observation.

Well, then, what is the grounding for belief in such values? Please explain how it follows from the actual attitudes of moral realists that there are independent moral values.