The following is based on my comments at the blog Tu Quoque.
I propose the following definition of the term "real":
A thing is real to the extent that one can settle propositions about that thing empirically (or computationally).In a logical positivist sense, all meaningful propositions are real (e.g., "456 + 341 = 797" is empirically verifiable). Strings of meaningless symbols are also real to the extent that we can state empirically testable propositions about those strings. Likewise, people and atoms are real to the extent that propositions about them are testable.
By this standard, even fiction has some level of reality. The fictional world of J.K. Rowling's Harry Potter is real, but only to the extent that we prefix our empirical propositions by "in the non-physical construction by J.K. Rowling..." We can create propositions about Hogwarts that can be empirically tested by verifying their consistency with Rowling's construction. However, outside of that specified literary context, Hogwarts is not real.
I think we can conclude that reality isn't a simple attribute like the truth value of a proposition. Instead, reality is a demarcation of context for meaningful propositions.
Two entities are part of the same reality when the very same empirical tests can be applied to both entities.I anticipate an objection to my claim based on a very narrow interpretation of what constitutes the "same empirical tests". The critic may ask, under my criteria, whether thunder and lightning are part of the same reality? I would answer that they are. Correlations of between empirical tests are themselves empirical. Using photodetectors and microphones, we can easily correlate thunder and lightning, but neither sensor will register anything when Hermione Granger casts a spell.
So, this is based on my definition of what the symbol "real" means. The key aspect of this definition is that it contains within it a relatively unambiguous recipe for determining whether something is real or not.
Now, you can dispute my definition. You may prefer a different definition, but my claim is that your definition is meaningless unless you have a recipe for deciding what is and what isn't real.
You see, the term "real" doesn't have a meaning until we define the term in a more precise fashion. I think much of the reason why metaphysics is so confusing to us is that we have a vague, gut, intuitive feeling for what "real" means. We then try to reason about what things are real based purely on the gut. Intuitively, it sounds like a good idea, but it's really a confusion.
Instead, let's create a set of rigorous definitions for what real means (real, real,... we'll call mine real), and then we can talk logically about what things are real.
Again, you may object to my definition on the grounds that there are things that you feel are real which do not conform to my definition. However, unless you have a rigorous alternative test procedure (real, say), your definition is just a gut feeling, and the definition of what is real gets reduced to how each of us personally feels about it.
As with the Principle of Verifiability, my statement about what is real is merely a definition. As such it is meaningful in the logical positivist sense as long as it specifies a procedure for determining reality.