I thought I might get some pushback on taking the word "real" in "real number" literally, because, as you say, real numbers are just as legitimate a mathematical object as anything else.

We probably differ, though, in how much we think of real & complex numbers as showing up in the real world. In practice, when I measure something quantitatively, the result's almost always a real number. If I count things I get natural numbers. If I can also count things backwards I get the integers. If I take a reading from a digital meter I get a rational number, and (classically) if I could look arbitrarily closely at the needle on an analogue meter I could read off real numbers.

But where do complex numbers pop up? To me they really only seem to inhere in quantum mechanics (where they are, admittedly, absolutely fundamental to the whole theory), but even there you have to work rather hard to directly measure something like the wavefunction's real & imaginary parts.

In the macroscopic world it's not easy to physically get at whatever complex numbers comprise a system's state. I can certainly theorize about the complex numbers embodied in a system after the fact; I learned how to use phasors in electronics, contour integration in complex analysis class, complex arguments to exponential functions to represent oscillations, and so on. But these often feel like mere computational gimmicks I deploy to simplify the mathematics, and even when using complex numbers feels completely natural in the exam room, the only numbers I see in the lab are real numbers.

As such I'm OK with informally differentiating between real numbers & complex numbers on the basis that I can point to any human-scale quantitative phenomenon, and say "real numbers are just right there", while the same isn't true of complex numbers. This isn't especially rigorous, but I thought that was a worthwhile way to avoid spending several introductory paragraphs trying to pin down real numbers more formally. (And I expect the kind of person who needs or wants an up-goer five description of real numbers would still get more out of my hand-waving than they'd get out of, say, "real numbers form the unique Archimedean complete totally ordered field (R,+,·,<) up to isomorphism".)