The measurement operators are matrices that come about as a result of assigning real eigenvalues (these are your possible measurement outcomes) to orthonormal vectors (your arbitrary coordinate system). The results are hermitian, complex-valued matrices, because that's just what comes out if you try to engineer a matrix to have those eigenvalues and vectors. The rest follows from that.
Trying to fit a real number constraint somewhere, other than the one that's already there (real measurement outcomes), to me seems like the step you would have to justify, not the absence of one.
The complex numbers in the matrices appear as a consequence of trying to do something else. I don't think they have much physical meaning on their own, which is why I am surprised that people ask what it would mean if they had to all be real numbers.
Trying to fit a real number constraint somewhere, other than the one that's already there (real measurement outcomes), to me seems like the step you would have to justify, not the absence of one.