In my field, we deal with data that are originally complex-valued. Typically, researchers convert their data from real + imaginary to magnitude + phase, and then discard the phase data (i.e., we generally only deal with the magnitude portion of a complex signal).
There are a number of papers on phase synchrony, in which the Hilbert transform is applied to this magnitude-only data to estimate the analytic signal, from which instantaneous phase is extracted. I am very much a novice in signal processing, so I found this a little confusing. The papers refer to the instantaneous phase signal estimated from magnitude-only data just as "phase signal". These folks might not be aware of the complex-valued nature of the raw signal, which is reasonable given how these data are generally handled, but I think there should be a more appropriate term for "phase estimated from magnitude signal using the Hilbert transform", given that there's very little similarity between the real phase signal and this estimated phase signal.
Is this an established concept and, if so, is there a term that would differentiate the two kinds of phase signal?