Perhaps no conceptual wall has been head-banged more than the eternal squabbling about the objective existence of "the real world". One way or the other, it simply cannot be proven whether there is something "out there".
I propose a simple solution, namely that non-existence itself does not apply to things.
Lets see step by step.
In the universe there are things, which are matter/energy, freely inter-convertible, but indestructible. You can take some atoms and zap them into energy and vice versa (someday). But it doesn't go away, even if it does(in a singularity), it's latent in the form of gravitational potential and can spring back into matter once more.
There are also labels, or concepts - Blue, Male, Infinity, Horrible and so on. These are words, or strings of bits which have an information-entropy within a context - that context is human civilization, history and the brain.
These patterns of bits can be messed up by expending energy, I can take a pot and pulverize it and remove the potness, I can burn a blue thing and make it black.
The labels exist and then are destroyed.
A similar thing is the set of pixels on this screen, where i can turn any number of them into any color including "black", which does not mean the pixel ceases to be.
Labels can be existent and non-existent, numbers can be infinite, living things can be male.
Concepts apply to things and to other concepts.
But, actual sets of physical objects can never have infinite cardinality, just like stones cannot be male or female. So also non-existence does not apply to indestructible matter/energy stuff.
The duality of the labels "existence" and "non-existence" only apply to the labels of objects (bit strings), and cannot apply to things, any more than you can classify a single atom as being liquid or gaseous.
Therefore, the world is..... and non-existence isn't.