$\vec E$ and $\vec B$ are in phase only in lossless media, for which the conductivity $\sigma=0$. In general, lossy media are modelled as having a complex permittivity $\epsilon$, which leads not only to an exponential decay of the amplitude but also to a phase shift between $\vec E$ and $\vec B$.

$\vec E$ and $\vec B$ are in phase only in lossless media, for which the conductivity $\sigma=0$. In general, lossy media are modelled as having a complex permittivity $\epsilon$: the complex part is proportional to $\sigma$ and this leads not only to an exponential decay of the amplitude but also to a phase shift between $\vec E$ and $\vec B$.