I've never thought of real numbers as any more real (in the non-mathematical sense of the word) than other numbers, and I've been peeved by popularizations which use “real” and “imaginary” without making it clear that they're using them with a specific technical meaning (e.g. stuff like “special relativity has shown that if space is real time must be imaginary, and vice versa” -- yeah, they do have squares with opposite signs (though modern notation uses real 4-vectors and a non-positive-definite metric), but that's not how a reader would be most likely to interpret that sentence).