Terminology for instantaneous phase of magnitude component of complex signal
and then discard the phase data

What's gone is gone. You can't reconstruct the phase of the original data unless you have some additional information.

in which the Hilbert transform is applied to this magnitude-only data to estimate the analytic signal, from which instantaneous phase is extracted.

That's a just mathematical process. You shove data in, and data comes out. Whether the result is useful or not depends on what you want to do with it and why you chose this mathematical process in the first place. If you want to associate any physical meaning with the result you need to justify this with some physical property or law.

To clarify, the Hilbert transform applied to the real component of a complex signal is not a good estimate of the original complex signal?

Correct. In general most signals are NOT analytic unless there is a strong reason why they should. If there is a reason to interpret the results that way, it should be stated in the paper. In order to &quot;estimate&quot; a discarded phase, you need have some additional information or (justified) assumptions about your original signal otherwise you might as well just toss the dice. If there is a additional information, it should be stated AND the phase estimation method should be properly derived from it.