Context: Refering to the question: Complexity of the $(3,2)_s$ SAT problem? and since the paper by Porshen and Speckenmayer : Satisfiability of mixed Horn formulas, we know that even when $F_3$ is Horn, the problem of deciding the satisfiability of $F_3 \wedge F_2$ is NP-complete - where $F_3$ and $F_2$ are respectively 3-CNF and 2-CNF formulas.
I am wondering if there exist some cases where $F_3 \wedge F_2$ is easy to decide. Hence my question:
Let $F_3$ a 3-CNF containing only clauses with exactly 3 different literals and $F_2$ a 2-CNF defined on the same variables as $F_3$.
What is the complexity of deciding the satisfiability of $F_3 \wedge F_2$ when $F_3$ and $F_2$ are both monotone?