I am looking for related work that matches a set-up as follows:
Consider a cocomplete category $\mathsf X$ (i.e. a category with all colimits), and a compatibility relation $\ast$ that is a functor $$ \ast : \mathsf X \times \mathsf X \to \mathsf X $$ such that $\ast$ is commutative (i.e. $x \ast x' = x' \ast x$; this may be on-the-nose or a natural transformation), and $\ast$ commutes with colimits on both components.
A simple example of this set-up is where $\mathsf X$ is a lattice, and $x\ast x'$ just returns $x\wedge x'$.
Other targets on the right-hand side are also possible, i.e. $$ \ast : \mathsf X \times \mathsf X \to\mathsf Y $$ where $\mathsf Y$ is a cocomplete category. Again to take a simple example, $\mathsf Y$ may be a lattice of truth-values, such as $\mathsf{Bool}$.
I welcome references to related work. Thank you.
This is essentially the notion of (symmetric) monoidally cocomplete category, which is a cocomplete symmetric monoidal category whose tensor product is cocontinuous in each argument. As mentioned in the comments, every cocomplete closed monoidal category is monoidally cocomplete, since in this case the tensor is left-adjoint and hence preserves arbitrary colimits.
For instance, see the introduction to Im–Kelly's A universal property of the convolution monoidal structure, which I believe is where the concept was first named.