The terminology regarding 'real' and 'imaginary' component of electrical (or magnetic) field

[jiajiasun]

I am getting confused by the terminology regarding 'real' and 'imaginary' components of the electrical (or magnetic) field. For plane waves in a homogeneous halfspace, the electrical field can be written as:
E = E_0 exp(-\alpha z) exp(-i (\beta z - \omega t)), which is obviously a complex number. Therefore, the real component is Real(E) = E_0 exp(-\alpha z) Real(exp(-i (\beta z - \omega t))) = E_0 exp(-\alpha z) cos(\beta z - \omega t). However, this real component is NOT in phase with the sheet current, I = I_x cos(wt), as shown on this webpage (https://em.geosci.xyz/content/maxwell1_fundamentals/harmonic_planewaves_homogeneous/derivation.html).

If we consider sinusoidal waveforms, and not using complex variables, the electrical field can be written as E = E_0 exp(-\alpha z) cos(\beta z - \omega t) = E_0 exp(-\alpha z) [cos(\beta z) cos( \omega t) + sin(\beta z) sin(\omega t)]. It then seems to me that the real (or in-phase) component of the electrical field should be E_0 exp(-\alpha z) cos(\beta z) cos( \omega t) (note that this is in phase with the sheet current), whereas the imaginary (or out-of-phase) componet should be E_0 exp(-\alpha z) sin(\beta z) sin( \omega t) (note that this is out of phase with the sheet current).

It seems that we cannot equate the real (and imaginary) concepts in the context of complex variables with the real (and imaginary) components of the EM field in the frequency domain.

I believe, most likely, I am wrong. Could anyone please point out where I went wrong. The real vs imaginary thing is really confusing me. Thanks a lot!