Sometimes it's useful to define more than one monoidal product on the same category. A good example is $\textbf{Set}$, where you have the product and the coproduct.
If we choose one or the other of those, we can use string diagrams, which are very useful. For $\textbf{Set}$ we can choose the product, in which case parallel boxes on parallel wires represent two functions being evaluated simultaneously on different arguments, or we can choose the coproduct, in which case the same diagram represents either one function being evaluated or the other.
However, sometimes it seems like it would be useful to be able to represent both monoidal operations at once somehow, for example, in a computational context in which we might have both branching and parallel execution.
Does there exist a diagrammatic calculus, similar to string diagrams, that would be suitable in such a case?
(I've kept the question deliberately vague, because I don't know what such a diagram might look like, or what additional axioms would have to be obeyed to make it work.)
Comfort–Delpeuch–Hedges introduced sheet diagrams for rig categories, i.e. categories with two monoidal structures, one of which distributes over the other, in Sheet diagrams for bimonoidal categories (2020), for which $(\mathbf{Set}, {\times}, {+})$ is a prototypical example.