Transcix, I'm using "real" in the Parmenidean sense, such that, if it isn't the infinite whence the finite comes - if it isn't you - it isn't real. It's perfectly apt to replace all labels with "illusion" - at least then you know you're looking at one, and not many! At that, if it's called an illusion, it is most certainly implied that it exists ("to be is to be perceived"), and yet is not real (just as my notion of Donkey Kong exists, though the character is not real).
It can be proven that the infinite can - indeed, does - rest apart from the finite, by examining the qualities of these kinds of existence. The finite is finite not only in form, but also temporally: it is born, it grows, it decays, it dies. Furthermore, one might develop words with which to describe the finite, and one might entertain notions of the finite. It is the qualities, the attributes, of the finite which allow us to label certain parts as distinct from others in the first place. One cannot do this with the infinite: though we point to it with that word, it is actually colourless, tasteless, formless, soundless, and so on. Attributeless, it is non-distinct, and indistinguishable. There is no quality to hold and say "yes, this is it" - the holder cannot hold itself. However, while one could not read words if there were no page, one could see the page without words on it.
Advaita Vedanta is specifically geared towards this kind of "self-discovery". The method is not to provide you with an experience, or an ideal, or any such thing that your mind might grab and hold onto in order to proclaim its own advancement. The method is to allow the infinite to express through the finite its own self-recognition in that manifestation. This is done through consistently dropping all things - "neti, neti", "not this, not that" - until one is no longer identified with any phenomenon. Not the body, not the mind, not the feelings, nor the world, nor anything that arises and ceases. What is left? Only the one who watches all of this, eternally untouched by what is occurring within itself. "The Kingdom of God comes with no signs to be perceived": this is how illusion and reality can be distinguished between, for the one is distinguishable, the other not. When all phenomena cease - say, when the body is dead, and no sensation remains - still there is the awareness of that lack of phenomena. In that timeless time, it is only I, alone, and no illusion. Clearly I exist apart from any phenomenon.