All practical probabilistic proof/argument systems I know of are based upon polynomial identity testing in finite fields. These constructions include QAPs, STARKs, GKR and many variants. In particular, they all rely on the Schwartz-Zippel lemma, and thus they all take place in a polynomial setting.
There are constructions that do not take place in a polynomial setting, such as the expander-graph based PCP. But they are impractical both asymptotically and in practice.
Are there constructions that do not take place in a polynomial setting, but are still potentially practical?