A complete $k$-CNF formula is a $k$-CNF formula which contains all clauses of size $k$ or lower it implies.
Deciding the satisfiability of a $k$-CNF formula is clearly a tractable problem since a $k$-CNF formula is satisfiable as long as it does not contain the empty clause. What happens when it is mixed with a 2-CNF formula?
Let define the Complete (3,2) SAT problem : Given $F_3$, a complete 3-CNF formula, and $F_2$, a (complete) 2-CNF formula ($F_3$ and $F_2$ are defined on the same variables). Is $F_3 \wedge F_2$ satisfiable?
What is the complexity of this problem ?
(The question is different as in the post Complexity of the (3,2)s SAT problem? where it concerned non complete formulas.)