But we have a common understanding of what "mostly empty space" looks like and what we perceive as solid objects don't look like it.

I think I can be more clear. When you refer to what something looks like you are referring to the way we perceive things, which is with electromagnetic waves. In a very real sense, with respect to that, (that is, with respect to the electromagnetic field), solid objects really aren't "full of empty space," they are chock full of electromagnetic fields. Like the idea of "empty" is actually just sort of not a great way of thinking about small stuff. In fact, if we want to operationalize our notion of full or empty, we'd probably end up back at our intuitive notion of the idea.

For example. We might say that a helium atom is "mostly empty space" because if we measured the position of one of the electrons it would tend to be far away from the other one (maybe the right way to think of this is to look at the spread of coherent position states). But, in fact, around a helium atom the electrons interact pretty strongly (enough to distort their wave functions substantially) so in that sense the region around a helium atom is significantly less empty than the space around the sun between (say) mercury and venus, which have very little effect on one another. What I am getting at is that all this idea of "atoms are mostly empty space" gets at is that our definition of "empty space" is pretty vague. Vague enough to do some silly word gymnastics.

Thanks! This is helpful. It seems like I took an overly-simplistic explanation of the small scale and took it too literally. There's still a nugget of truth to the idea that our perceptions only partially tell us what the universe is really like, but it's not as disturbing as my initial thought that I was somehow observing something ghost-like as if it were solid.

Solid objects look exactly like what we perceive solid objects to look like. That is kind of the point.