  We discuss the modified quantum electrodynamics from a time-reversal-breaking
Weyl semimetal coupled with a $U(1)$ gauge (electromagnetic) field. A key role
is played by the soft dispersion of the photons in a particular direction, say
$\hat{z}$, due to the Hall conductivity of the Weyl semimetal. Due to the soft
photon, the fermion velocity in $\hat{z}$ is logarithmically reduced under
renormalization group flow, together with the fine structure constant.
Meanwhile, fermions acquire a finite lifetime from spontaneous emission of the
soft photon, namely the Cherenkov radiation. At low energy $E$, the inverse of
the fermion lifetime scales as $\tau^{-1}\sim E/{\rm PolyLog}(E)$. Therefore,
even though fermion quasiparticles are eventually well-defined at very low
energy, over a wide intermediate energy window the Weyl semimetal behaves like
a marginal Fermi liquid. Phenomenologically, our results are more relevant for
emergent Weyl semimetals, where the fermions and photons all emerge from
strongly correlated lattice systems. Possible experimental implications are
discussed.
