What is tropical field theory?

Tropical field theories are a new class of model quantum field theories which retain most of the combinatorial and algebraic features of perturbative quantum field theory, while drastically simplifying the numerics. A tropical field theory can be defined purely combinatorially, or equivalently it arises as the tropical limit of parametric Feynman integrals. From the physicist's perspective, tropical field theory is a special case of long-range interacting field theories.
I will use tropical scalar $phi^4$ theory as an example to explain the equivalent definitions, and to illustrate what can be learned about renormalized quantum field theory from such a tropical model.