And so it seems to many realists as well. For instance Karl Popper argued for the reality (the real existence) of ideas.
The problem is where we assert, construct or discover a correspondence between an abstract idea and a real object, and then give the former precedence over the latter. The real must always be the test of the ideal, not the other way around. Furthermore, even if both object and idea are real, the correspondence between idea and object is not real (as Plato argues) but only incidental and instrumental. The correspondence is a kind of "theory" subject to testing and to refutation.
We can't follow this advice until we distinguish between the conceptual and the actual ideal.