Interpretation of the Platonic forms is a big wrangle and may be undertaken in many ways. But the simplest answer to your question "in what realm" might just be to say everywhere or in every "realm."
Thus the "squareness" of a square, to use the old trope, is not "in" this or that particular square thing. In fact, any particular square will not be perfectly square and will not stay square forever. Yet we may see "squareness" reflected in this or that thing and so attach the geometrical predicate. To Plato this is not mere immanence, for it indicates that "squareness" is more real than any things or brain states dimly reflecting it.
The view goes back to Parmenides that what does not change cannot, of course, be itself perceived by our changing sensory input, but is for that very reason more real than shifting phenomena, thus cannot depend upon those phenomena and so exists prior to and independent of any such phenomena.
These are not easy adjustments for modern minds, but not a few notable mathematicians and physicists, Goedel and Penrose, for example, are Platonists in this sense. They simply cannot feel they are "making things up" mathematically or arriving at solutions by way of observations. They feel they are struggling beyond perceptions to discover something objectively "out there."
Perhaps another way to think about it is to consider gravity. Where is it? Everywhere. What is it? It is not this or that material object in which we see "gravity" reflected. It is basically a mathematical form of the universe that we do come to think of as very real, though it would be problematic to say it is "in" the universe.
We are also thrown off by the word mathematical "object," which suggests perceptible things. Gravity or squareness or oneness can be "objects" of our consideration unseen but for, as we say, the practiced "mind's eye," just as a radio antenna is needed to hear the voices "really" in the airwaves all around you. The mind "senses" not a thing but a limit or necessity that we "know must be the case" and so must "be."
To Pythagoreans like Plato such was the "real" necessity of numbers and forms, without which no "ideas" could even begin to take shape out the "radio static" of unformed sense data.