Given two propositional formulas over a set of literals with AND, OR, and NEGATION, find propositional formulas $\phi_1$, $\phi_2$, and $\theta$, s.t. $\phi_i = \theta \wedge \phi_i^*, \; i=1,2$, and $ (\phi_1^*, \phi_2^*, \theta) = \underset{\phi_1^*, \phi_2^*, \theta}{\operatorname{argmin}} (|\phi_1^*| + |\phi_2^*|+ |\theta|)$.
The minimum itself is upper bound by $|\phi_1| + |\phi_2|$ setting $\theta=True$. As far as I can see, the solution should be easy and computable without NEGATION, by computing the minimal positive CNF and just factoring out common disjunctions.
But is there also a solution in this more general case and how to compute it?