On the union stabilization of two Heegaard splittings

Let two Heegaard splittings $V_1 \cup W_1$ and $V_2 \cup W_2$ of a 3-manifold $M$ be given. We consider the union stabilization $M=V \cup W$ which is a common stabilization of $V_1 \cup W_1$ and $V_2 \cup W_2$ having the property that $V=V_1 \cup V_2$. We show that any two Heegaard splittings of a 3-manifold have a union stabilization. We also give some examples with numerical bounds on the minimal genus of union stabilization. On the other hand, we give an example of a candidate for which the minimal genus of union stabilization is strictly larger than the usual stable genus -- the minimal genus of common stabilization.