  The solution of the Dirac equation in the presence of an arbitrary plane
wave, corresponding to the so-called Volkov states, has provided an enormous
insight in strong-field QED. In [Phys. Rev. A \textbf{103}, 076011 (2021)] a
new "fully quasiclassical" representation of the Volkov states has been found,
which is equivalent to the one known in the literature but which more
transparently shows the quasiclassical nature of the quantum dynamics of an
electron in a plane-wave field. Here, we derive the corresponding expression of
the propagator by constructing it using the fully quasiclassical form of the
Volkov states. The found expression allows one, together with the fully
quasiclassical expression of the Volkov states, to compute probabilities in
strong-field QED in an intense plane wave by manipulating only 2-by-2 rather
than 4-by-4 Dirac matrices as in the usual approach. Moreover, apart from the
exponential functions featuring the classical action of an electron in a plane
wave, the fully quasiclassical Volkov propagator depends only on the electron
kinetic four-momentum in the plane wave, which is a gauge-invariant quantity.
Finally, we also compute the one-loop tadpole diagram in a plane wave starting
from the Volkov propagator and we find that after renormalization it
identically vanishes.
