We have to be careful when using terms like "real world" and so on. There is a propensity to associate "real" with "ultimate reality", permanence, eternity, heaven etc. As though describing "places". Like you can be "in" reality or outside of it. If something is considered real, you might equate it with Heaven for example.
But this is not really what ACIM means when it refers to things as real. For example it tells you to "make the world real unto yourself", while in the next breath "there is no world".
Real, in these terms, really means "aligned with truth". Or let's say a person is insane, and to that insane person they experience things in an unreal way, a distorted way. When they return to sanity, they return to "reality". In other words, to "what is", or the truth about what is here, rather than the distorted view of it.
So the world we made, Earth, can "become real to you", in the sense that it it can be seen with Christ Vision from a perspective of real love, which lends it the meaning of love. Or moreso, truth. And so now the world is seen in the light of truth and for what it "really is". But this doesn't mean it's real as in ultimate reality. It means you're having what Jesus describes as "accurate perception" of it.
That accuracy, without distortion, is what it means when he refers to "the real world". A world as seen through loving eyes that reveals the "true nature of" the world. The true nature of the world, however, is that it is still an illusion of a world.
So when you make it real unto yourself, that means to love the world regardless of it being an illusion. To see it "with reality" or aligned with truth. Rather than seeing the "unreal world" which is seen through false perception, a distorted, condemned, dark version of the world. A misinterpreted world.
This is why "the real world" is called the real world. It doesn't mean it is reality as in, the location of Heaven. It doesn't mean it is "where ultimate reality is" either. But it does mean this is a state of mind and awareness in which the world is seen through vision, through sane eyes, recognized for what it actually is, rather than having bought into lies, illusions, deceptions or temptations. A clear mind, a right mind, looking with sanity upon simply what it is and nothing more or less.
We reach "the real world" through this process of sorting out perception and aligning with truth. Becoming able to see, without ego distortions and interruptions, what is actually there. What is there, as the world, without our projections of sin and guilt upon it. A forgiven world. A world seen through love's eyes. And in fact a world that God loves. BUT, this does not "make it real" in the sense of it being now some kind of permanent eternal reality. It just means it is now seen in the light, seen with truth, seen with sanity, and recognized. Its "true nature" or "the reality of it" is revealed. But the world's true nature is still that it is an illusion.
This is why "the real world still is but a dream". We move on from the real world.... the forgiven world... the world seen through "realities eyes" so to speak, to go "to reality", where we can now rightfully associate reality and location, or reality and permanence, or reality and eternal life. The Kingdom of God, where everything IS real, is recognized, and ALSO is eternal.
False perception is an illusory world seen in an "unreal way", which makes it seem like hell on earth.
The real world is an illusory world seen in a "real way". This makes it seem like heaven on earth.
Heaven is a real world, the Kingdom, seen in a real way.