Nominalism and realism have been a frequent subject in my online conversations of late. I think it would be beneficial to define the terms so that we are clear about what we mean. Primarily, I am concerned about realism and nominalism of universals and names. This statement may say more than it appears, because much of contemporary philosophy tends to equivocate on the term "realism."
There is a realism of the external world and realism of universals/names. The former is usually expressed in terms of a correspondence of idea or experience and what is experienced, which is invoked when we say that an idea of word has a real referent. The realism of universals/names is often thought in such ideas as Kripke's "rigid designation," although that is a cousin many times removed from the topic that I wish to discuss. In Kripke's case, a rigid designator is a name that refers to the same object in all possible worlds. It is a logical construction.
In contrast, the realism of universals/names argues for the reality of universals, e.g. the phenomenal qualities (white, red, vanilla, etc.), etc. that can be predicated of a subject. It is more an issue of onto-logy rather than mere logic, the logic of being rather than of signs. These two realisms are separable, but I hold them together, though not in the conventional way, though that is another topic. E.g., a felt quality is not merely a one-to-one mirroring of what is in the external thing; it is neither one-to-one nor a mirroring, which is what most contemporary realists (external world) would wish either in fact or ideally.
What's the point? Without a realism of universals, of which the phenomenal qualities are a case, then any experience becomes arbitrary. We run into all the problems of empiricism that Hume and Berkeley exposed. Should we then seek shelter in Kant and psychologism? Qualities are law-like by-products of human experience that have no basis in external reality? No.
What is at stake? Without a reality of universals, then phenomenology really is every bit of trash that most analytics think it is. This is in part why they almost universally denigrate it--because they are NOT REALISTS about universals, and thus they think that mere experience is hokum. And thus they retreat into a neo-Cartesian position of thinking that what is really real is the rational and intellectual, err … "scientific." This was part of Husserl's ferocious critique in the Crisis; they mathematized being and did not even realize it.
Finally, I have already indicated what is at stake in pragmatic process metaphysics elsewhere, which includes pragmatic phenomenologies. The latter are non-Husserlian and realist in a more robust sense than Husserl, and are scientific without being any flavor of scientific realist. What is at stake is the incoherence of a process metaphysics that rejects autonomous identity (substance) and becomes nominalist (what a thing is said to be, is arbitrary).