The definition of the relativistic Action is
$$ S=\int_a^b ds $$
The Lorentz invariant of electromagnetism is
$$ s^2=\frac{1}{c^2}||\mathbf{E}||^2-||\mathbf{B}||^2-2i\frac{1}{c}(\mathbf{B}\cdot \mathbf{E}) $$
Therefore, could we write the action as:
$$ S=\int_a^b d\sqrt{\frac{1}{c^2}||\mathbf{E}||^2-||\mathbf{B}||^2-2i\frac{1}{c}(\mathbf{B}\cdot \mathbf{E})} $$
This may not be the prettiest way to write it I concede, but is it possible? Clearly, this is not the field description, but perhaps the movement of a test particle/photon in an electromagnetic field?