A series of works has established rewriting as an essential tool in order to prove coherence properties of algebraic structures, such as MacLane's coherence theorem for monoidal categories, based on the observation that, under reasonable assumptions, confluence diagrams for critical pairs provide the required coherence axioms. We are interested here in extending this approach simultaneously in two directions. Firstly, we want to take into account situations where coherence is partial, in the sense that it only applies to a subset of the structural morphisms. Secondly, we are interested in structures which are cartesian in the sense that variables can be duplicated or erased. We develop theorems and rewriting techniques in order to achieve this, first in the setting of abstract rewriting systems, and then extend them to term rewriting systems, suitably generalized to take coherence into account. As an illustration of our results, we explain how to recover the coherence theorems for monoidal and symmetric monoidal categories.