The issue of numbers has been brought up. Now, there is a category of numbers called 'real numbers', but that is a terminological thing and not directly related to the issue of the existence of mathematical objects.
Before I would address the issue of numbers, though, I might ask if the game of chess is 'real'. Does it exist? The reason I ask is that it seems to me (as a mathematical formalist) that the existence of numbers is *exactly* like the existence of the game of chess.
In another line, is the *idea* of a unicorn 'real'?
What I would say is that the real/unreal distinction is primarily made to separate out those things that exist mainly because we think about them as opposed to things, like chairs, that exist whether we are thinking about them or not.
So, in this sense, the game of chess isn't 'real' because it primarily exists because we think about it. Similarly for unicorns. The *idea* of unicorns might exist if we can detect what is going on in the brain of someone who is thinking it. Electrons are real (yes, even in the double slit experiment) because it isn't primarily because of our thinking about them that they exist. They exist outside of anyone's consciousness. Abstractions are not 'real' in this sense either: it is our classification and our thoughts that determine whether and how they exist.
And no, I would also not say that numbers are 'real' (although, again, there are 'real numbers') in the most relevant sense.