Here's some quotes explaining why what ACIM terms as "the real world" IS NOT Heaven - they are two entirely difference "places" or experiences. The real world is firmly rooted in perception, Heaven is beyond perception, beyond Earth, beyond this universe, beyond bodies of any kind, and can only be known and experiences in real awareness. This is why "Heaven is an AWARENESS (not perception) of perfect Oneness".
You are to reach the "real world" first and then God will scoop you up into Heaven. But the real world, itself, is actually NOT REALITY, and is UNREAL. This terminology has confused many people. It merely means, "the world AS IT IS" (the "real" version of it) rather than your perceptual distortion about it. It's "relative" terminology, not absolute.
Also Heaven is still very DIFFERENT to Earth, due to the fact that even the real world and Earth are still illusions based on separation.
"THE REAL WORLD STILL IS BUT A DREAM. Except the figures have been changed. They are not seen as idols which betray. IT IS A DREAM in which no-one is used to substitute for something else, or interposed between the thoughts the mind conceives and what it sees. No-one is used for something he is not, for childish things have all been put away. And what was once a dream of judgment now has changed into A DREAM where all is joy, because that is the PURPOSE that it has. Only forgiving dreams can enter here, for time is almost over. And the forms which enter in the dreams are now perceived as brothers, not in judgment, but in love."
"Perception of the truth is not the same as KNOWING it." <- real world is PERCEIVED, and perception is NOT the same as knowledge. Knowledge is only of heaven, therefore the "real world" IS NOT heaven.
"For true perception is a remedy with many names. Forgiveness, salvation, Atonement, true perception, all are one. They are the one beginning, with the end to lead to oneness far beyond themselves." <- true perception as in the real world, POINTS TO something beyond itself.
"The world stands like a block before Christ’s face. But true perception looks on it as nothing more than just a fragile veil, so easily dispelled that it can last no longer than an instant. It is seen at last for only what it is. And now it cannot fail to disappear, for now there is an empty place made clean and ready. Where destruction was perceived the face of Christ appears, and in that instant is the world forgot, with time forever ended as the world spins into nothingness from where it came." <- in the real world true perception shows you that this world is just a fragile veil, easily dispelled - it's not reality or heaven, AND it disappears. The real world disappears.
"A world forgiven cannot last. It was the home of bodies. But forgiveness looks past bodies. This is its holiness; this is how it heals. The world of bodies is the world of sin, for only if there were a body is sin possible." <- forgiven world (real world) CANNOT LAST. And IT is the home of bodies, which cannot last either.
"Gone is perception, false and true alike. Gone is forgiveness, for its task is done. And gone are bodies in the blazing light upon the altar to the Son of God." <- no bodies in the light of God
"For where God’s memory has come at last there is no journey, no belief in sin, no walls, NO BODIES" <- no bodies in heaven, in any way
"The Holy Spirit has no need of time when it has served His purpose. Now He waits but that one instant more for God to take His final step, and time has disappeared, taking perception with it as it goes, and leaving but the Truth to be Itself." <- perception ends, ending the "real world" with it.
"The real world is a symbol, like the rest of what perception offers." <- the real world is PART OF PERCEPTION, AND PERCEPTION WILL END, so the "real world" will end also.
"For, as your sleeping and your waking dreams represent the same wishes in YOUR mind, so do the real world and the truth of Heaven join in the Will of God. The dream of waking is easily transferred to its reality." <- the "real world" is "the dream of waking".
Here is where you move BEYOND THE REAL WORLD:
"Here (in the real world) are YOU forgiven, for here you have forgiven everyone. Here is the new PERCEPTION, where everything is bright and shining with innocence, washed in the waters of forgiveness, and cleansed of every evil thought you laid upon it. Here there is no attack upon the Son of God, and YOU are welcome. Here is your innocence, waiting to clothe you and protect you, and make you READY FOR the final step (to heaven) in the journey inward. Here are the dark and heavy garments of guilt laid by, and gently replaced by purity and joy.
Yet even forgiveness is NOT THE END. Forgiveness DOES make lovely, but it does NOT create. It IS the source of healing, but it is the MESSENGER of love, and NOTits Source. Here you are led, that God Himself can take the FINAL STEP unhindered. For here does nothing INTERFERE with love, letting it be Itself. A step BEYOND this holy place of forgiveness (beyond the real world). A step still FURTHER inward, but one you CANNOT take, transports you to something COMPLETELY DIFFERENT (heaven). Here is the SOURCE OF LIGHT; NOTHING PERCEIVED, forgiven, nor transformed. But merely KNOWN." (heaven).
"Heaven remains your one alternative to this strange WORLD *YOU* MADE and all its ways; its shifting patterns and uncertain goals, its painful pleasures and its tragic joys. God made no contradictions. What denies its own existence and attacks itself is not of Him. He did not make two minds, with Heaven as the glad effect of one, and earth the other's sorry outcome which is Heaven's opposite in every way." <- even the real world is STILL opposite to heaven in every way. It it seen WITH love but is NOT love. Heaven is ANOTHER world beyond the real world.
"There walks with you the Spirit Heaven sent you that you might approach this door some day, and through His aid slip effortlessly past it to the light. Today that day has come. Today God keeps His ancient promise to His holy Son, as does His Son remember his to Him. This is a day of gladness, for we come to the appointed time and place where you will find the goal of all your searching here and all the seeking of the world, WHICH ENDS together as you pass beyond the door." <- speaking of going beyond the real world to heaven, from "no one can fail who seeks to reach the truth." lesson 131
Entry to heaven: "Put out your hand and see how easily the door swings open with your one intent to go beyond it. Angels light the way, so that all darkness vanishes and you are standing in a light so bright and clear that you can understand all things you see. A tiny moment of surprise, perhaps, will make you pause before you realize the world you see before you in the light reflects the truth you knew, and did not quite forget in wandering away in dreams."