As I understand it, nothing that can be changed, or broken down into smaller parts is inherently real.
That's not inaccurate, but it is a dangerous way to put it, if we don't qualify the terms.
"Inherently real" here has a specific meaning--it means "existing from its own side" (as the Tibetans put it), or, in Western terminology, a substance. As Spinoza put it, "By 'Substance', I understand that which exists in itself, and is conceived by itself, i.e., that which does not need the conception of any other thing in order to be conceived."
So, a substance inherently exists in that its existence is not dependent upon any other thing.
So, the cup on my desk does not inherently exists, as its existence was dependent upon a factory making it, etc.
If my worldly perceptions of emptiness are not inherently real, but created out of some interaction between emptiness and my own mind, how does that explain that so many perceptions are shared?
First off, let's be careful not to reify "emptiness"; emptiness is itself empty. So, my worldly perception of the cup is not the effect of the interaction between my mind and "emptiness" per se, but between my mind (which is empty) and the cup, (which is empty).
No one is saying the that cup isn't there, or that I am imagining it, or anything like that. Solipsism doesn't raise its head. The point is that the cup is not existing in and of itself, partaking of an eternal Platonic essence of cupness, without dependence upon any other object.
There are a number of good English-language books on Madhyamaka which are aimed at readers with a background in Western philosophy; I'd particularly recommend Jan Westerhoff's "Nagarjuna's Madhyamaka", Jay Garfield's "The Fundamental Wisdom of the Middle Way" (which is a translation with commentary of the MMK), and Mark Siderits's "Buddhism As Philosophy" (which is an overview of Buddhist philosophy tout court, but has a chapter on Madhyamaka and emptiness.)