One possible replacement is the computable numbers [1]; this includes the algebraic numbers and some common transcendentals (e, pi), and you can even build up something akin to standard analysis (computable analysis [2]).
The trick here is defining "occur" and "real world" precisely. Are you saying the act of me writing down those symbols and expressing the idea does not count as "occurring in the real world?" ;)
This reminds me of the self-defeating property of an "uninteresting" number -- a reasonable definition might be "any number that does not have any property of human interest", but then of course there is a smallest such number, and so that has the interesting property of being the first uninteresting number, a contradiction!
You're right though that a more precise definition of "real-world numbers" is needed, but I confess that my attempts to think of one in the past few minutes have been essentially circular (coming down to "the ones we know how to compute")!
It's not clear whether the universe is computable, however, in the sense that we only find computable numbers in nature. This is kind of an epistemological catch-22, though. How would we know whether this were the case or not?