Great question. Obviously, without explicit notes on the subject from the author, this is a judgment call. However, before I share my thoughts on it, I want to make a distinction. Remember Plato's allegory of the cave? In it his philosopher (unchained man) saw the truth (the sun), but when he tried to explain it to follow citizens (those still in chains), he could only explain it in terms of things familiar to them. It was real, but they took it as a story. This fits with his general idea that the realm of ideals is more real than the world of the senses in which we live.
So, is the Republic something he wanted realized? It is already real, and so the question changes a bit. However, that said, if pressed to answer yes or no, I'd say he wants it to spur discussion and thereby nudge society closer to it being real.