Third Way to Interacting $p$-Form Theories

We construct a class of interacting ($d-2$)-form theories in $d$ dimensions that are "third-way" consistent. This refers to the fact that the interaction terms in the $p$-form field equations of motion neither come from the variation of an action nor are they off-shell conserved on their own. Nevertheless, the full equation is still on-shell consistent. Various generalizations, e.g., coupling them to ($d-3$)-forms, where three algebras play a prominent role, are also discussed. The method to construct these models also easily recovers the modified three-dimensional Yang-Mills theory obtained earlier and straightforwardly allows for higher derivative extensions.