"The real world still is but a dream." - section 7 of Chapter 29, section 4 of ACIM.
In the Course, sometimes Jesus uses some terms which SEEM to imply something He did not mean to imply. When he referred to the "real world", He did NOT mean the world is real. And by world here, I'm talking about the physical universe, because every reference to "world" in the Course is referring to the physical universe (the dream of death).
So Jesus referred to the world, seen correctly with "true perception", ie perception without any errors - still perceiving WITHIN the dream, within space and time, but without any delusions ... as the "real world". It is "real" as in, seen WITH TRUTH, the truth OF it is seen correctly. The truth OF it is that it is NOT REAL. The truth of it, is that it is FALSE and untrue. When this is RECOGNIZED with true perception, you are seeing the "real version" of the world. In other words, real as in, what it actually is (unreal) compared to what you thought it was (illusions appearing real).
He uses this terminology because He's comparing it to when you have false perception, where you see this physical world with eyes of hate and judgement, and do not recognize it for what it is. In false perception you see a world of vengeance. You see hate IN the world. When you compare that false view OF the world, with a CORRECT view OF the world, the correct view is seeing "the truth" ABOUT the world... which shows you, in correct recognition, that this world is FALSE and NOT REAL.
The world is still an illusion. A false perception OF an illusion makes it seem real to you, and then you do not recognize it is an illusion, and you think it is the truth. That mistaken perception of the world needs to be corrected in order to see the FORGIVEN WORLD. When the world is seen through forgiving eyes - eyes of truth - those eyes do not see hate and guilt and fear IN the world. It simply sees what is "there", which is... an illusory world. It SEES and recognizes that there is a FALSE WORLD appearing in perception. You might well see God "in it", or see God "about it", or overlook the world to see Christ beyond it. But the world ITSELF is recognized as NOT REAL. The "real world" is actually the NOT REAL WORLD.
Once the world is recognized as not real, ie you are finally able to SEE what this thing IS for what it actually is - a false illusion pretending to be a world - then the purpose of perception and of the world is complete, and THEN YOU WAKE UP FROM THE DREAM, the world disappears, and there is only you with God. NOW you are in reality.
Jesus's use of the term "real world" in the Course has been very confusing for many students, because so many of us rely on the exact specific words Jesus uses and we don't really take it all in proper context of the bigger picture. This creates contradictions. If Jesus really meant the "real world" was actually REALITY, this would go against 99% of the rest of the Course. He ONLY uses this term to describe a correct perception of an UNREAL world - see the "reality" or "truth" of the world, which shows you that it is not really there. It's not a real world at all.
Jesus actually does this also in some other places in the course where he uses a terminology that SEEMS contradictory or seems to suggest something He did not intend. To discern this you need quite a bit of experience grasping the metaphysics. For example He said in one place that something can be "perfectly real within time but not in eternity." That can seem to imply that something inside of time - even though the course teaches that time is an illusion - is "real". If you interpret it that way, you will now come to a vast array of further confused conclusions. Jesus is not saying that anything in time is real at all. He is only saying that eternity is real and anything in time is an ILLUSION of reality, and seeing it correctly is seeing the "reality of it" - the truth that it is an illusion, just as above.
The "real world" is not real. There is no real world. This is the central thought the course would attempt to teach. This is the thought which anyone with an ego DOES NOT WANT. The fact is that when you get to True Perception, you WILL and MUST recognize that this thing you've been calling a real world, is JUST an illusion. Nothing more or less. It has no reality to it, no truth to it, it is recognized and registers with you as completely FALSE. You see THROUGH it, with forgiving eyes. You do not buy its illusions, you do not believe in it, you do not think it is really happening. And then it disappears as God scoops you up and you wind up back in REALITY. A reality in which THE WORLD DOES NOT EXIST. Or in a word, "Heaven". Only heaven is real.
"THE WORLD YOU SEE DOES NOT EXIST" (the world you see with your physical eyes does not exist).
"Forgiving dreams have little need to last. They are not made to separate the mind from what it thinks. They do not seek to prove the dream is being dreamed by someone else. And in these dreams a melody is heard that everyone remembers, though he has not heard it since before all time began. Forgiveness, once complete, brings timelessness so close the song of Heaven can be heard, not with the ears, but with the holiness that never left the altar that abides forever deep within the Son of God. And when he hears this song again, he knows he never heard it not. And where is time, when dreams of judgment have been put away?" - ACIM