One way to explore the idea of "interacting fields" in classical
physics would be do see if Faraday and Kerr effects, discovered
in 19th century, can be observed in a vacuum. Send polarized light
through static magnetic, or static electric, field and see if the
orientation of the plane of polarization changes. If it does then
the idea of "interacting fields" must be accepted.
My prediction is that the outcome would be negative, static fields
act on atoms and atoms, in turn, act on light. But who knows,
perhaps something remains when most atoms are removed. I assume
that this has already been investigated, long time ago. Does
anybody know a reference?