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How do we actually visualize the effect of two fields interacting in the same region of space?

If fields are just mathematical formulations to explain things that have no physical meaning, how are field interactions formulated and studied theoretically that can explain practical observations?

First you suggest fields are somewhere specific, raising the question of how they can share locations. Then you suggest fields don't describe real physics, raising the question of how they can exhibit cause and effect. Neither way of thinking about fields is right. They're functions, and each has a value everywhere. Just as $x^2$ doesn't have a location, nor does $A_mu$; just as $f^2+g^2=1,,f=g'$ constrain two functions, so field equations constrain the functions we call fields.

Good question.

Almost all practical and scientific progress in high energy physics in the last decades has been related to lattice regularization of field theories.

(If you have access to it, see Kogut's article)

Not only has this direction brought about genuine accurate predictions in high energy physics but it provides an intuitive connection to statistical mechanics via the Feynman path integral. In this framework fields live on sites, links, etc... of a space-time lattice. Interactions can be seen with your eyes plain as day in terms like: $phi^dagger_x U_x, mu phi_x+mu$, which shows an interaction between two fields on neighboring sites ($phi_x$ and $phi_x+mu$) via a gauge field on the connecting link ($U_x,mu$).

I would say if you are interested in making contact with reality in field theory to look into ``lattice field theory'' and start reading papers from the late '70s.

As for whether such a model has ``no physical meaning'', this is a question of philosophy, and you can ask it on SE philosophy, but keep in mind the practical quality of holding a belief that your successful model is actually a reflection of reality: you gain intuition about how the natural world interacts and behaves and as such you may be able to even improve your understanding and make progress in explaining un-explained phenomena.