Well, that depends on what you mean by “really exist.” The word “real” has a number of possible meanings, and with some of those the question is meaningless, but not others.
For example, I can say that silk or plastic flowers aren’t “real flowers.” That’s not meaningless at all. There is a viable distinction to be drawn between genuine flowers and artistic replicas of flowers.
If I “see” something out of the corner of my eye when there is nothing there, then it’s also meaningful to say that what I saw wasn’t “real.” I thought it was one thing, but actually it was another. Similarly, if I imagine that someone is expressing anger at me, when they’re really angry at someone or something else, then it’s meaningful to say that their anger at me is not real.
Where things get tricky is when we try to apply similar thoughts to the universe as a whole, or to all of the world perceived by the senses. Is that world “real”? Well, yes, in one sense – we experience it, therefore it is real. Duh! And we can also make predictions about how it will behave, using cognitive modeling. That systematic behavior is also “real,” because we observe it.
But it is one thing to say that something is “real” because it is observed or experienced, and quite another to say that something is “real” even outside of being observed or experienced. How in the world would we ever verify that?
Looking out my window, I see a tree. I also saw a tree that looked much the same yesterday. I expect, with much confidence, that I will also see a tree that looks much the same if I look out the same window tomorrow. I can say, using shorthand, that “there’s a tree outside my window.” What I mean is: “If I, or anyone else, looks out this window, they will see a tree.” The act of looking out the window (or looking from some other perspective, or feeling the bark, or climbing, or smelling the leaves, or hearing the leaves rustle in the wind) is a crucial part of what I mean by saying “there’s a tree outside my window.”
If I say, “There’s a tree outside my window, and it is there even if I never saw it and nobody else did either,” then I’m saying – what, exactly? I can’t be talking about looking at the tree, or feeling it, or hearing it. And in what other mode does the tree exist? Thinking about it, maybe. So the only meaning the above statement has is, “One may think about a tree outside my window that one has never seen.”
Reality exists for us in four modes: sensation, imagination, cognition, emotion. When we say that something exists, we mean we can sense it (see it, hear it, touch it, smell it, taste it), or imagine it, or think rationally about it, or have feelings about it. And that is all we mean.
The “objective universe” appears to which of these functions? Mostly cognitive thought, and to a lesser degree, emotion. We can’t sense the “objective universe,” nor can we imagine it (since that requires imagining a sensory experience), but we can think cognitively about it – so that it what “objective reality” is: a cognitive structure in the mind, used to model the world of sensation.