We characterize all finitary consequence relations over $\mathbf{S4.3}$, both syntactically, by exhibiting so-called (admissible) passive rules that extend the given logic, and semantically, by providing suitable strongly adequate classes of algebras. This is achieved by applying an earlier result stating that a modal logic $L$ extending $\mathbf{S4}$ has projective unification if and only if $L$ contains $\mathbf{S4.3}$. In particular, we show that these consequence relations enjoy the strong finite model property, and are finitely based. In this way, we extend the known results by Bull and Fine, from logics, to consequence relations. We also show that the lattice of consequence relations over $\mathbf{S4.3}$ (the lattice of quasivarieties of $\mathbf{S4.3}$-algebras) is countable and distributive and it forms a Heyting algebra.