When to use (and when not to use) electromagnetic field conjugates in variational formulations

I found something a little bit confusing about writing variational formulas or Lagrangians for electromagnetic fields. I was looking at the book by Schwinger and Milton (chapter 4), and saw that initially they only worked in terms of time-harmonic $E$ and $H$ fields without any conjugates ($E^{*},H^{*}$) appearing in the Lagrangian. This is a bit confusing because, from traditional treatments of Poynting’s theory in time-harmonic field, we are used to seeing complex conjugates appearing in second order terms all the time, and the Lagrangian should (in principle, I think) have some of those same energy density terms. For example, in their eq (4.13) they give a Lagrangian of the form:

$$ L=int_{V} dV left( frac{1}{2}iomegamu H^{2} -frac{1}{2}iomegaepsilon E^{2} – mathbf{E}cdotnablatimesmathbf{H} +frac{1}{2}sigma E^{2} +mathbf{J}cdotmathbf{E} right)$$
where taking $delta H$, for example, while requiring $delta L$ to be stationary, will lead to retrieving Maxwell’s equations. This formulation doesn’t not show any conjugates.

However, they start to use conjugates $E, E^{*}, H, H^{*}$ in the formulation when the medium is dissipationless, but without much explanation on why that is the case. Further, they mention that $E$ must then be treated independently from $E^{*}$, and $H$ from $H^{*}$, but I am not sure why. Their eq (4.29), for example, gives the Lagrangian for a dissipationless system ($epsilon, mu$ areal, and $sigma=0$) as
$$ L=int_{V} dV left( -iomegamu mathbf{H}cdotmathbf{H}^{*} -iomegaepsilon mathbf{E}cdotmathbf{E}^{*} + mathbf{H}^{*}cdotnablatimesmathbf{E}-mathbf{H}cdotnablatimesmathbf{E}^{*} +mathbf{J}cdotmathbf{E}^{*}-mathbf{J}^{*}cdotmathbf{E} right)$$
which is now varied in terms of $delta E$, $delta E^{*}$, $delta H$, $delta H^{*}$ as four independent cases, to retrieve Maxwell’s equations and their conjugates!

1. When do we use the conjugate fields in the variational formulation and when do we not? And how do we compare this with terms that appear in Poynting’s theorem for time-harmonic fields, which contains conjugates?
2. When is a field (say $mathbf{E}$) independent from its conjugate (say $mathbf{E}^{*}$), and when is it not?
3. And how is that related to dissipation in the medium (for example if we take the permittivity $epsilon$ complex, instead of real)?