 Wedge positivity for two positive (1, 1)-forms

If $\phi$ and $\psi$ are positive real $(1,1)$-forms and $\xi ,\eta$ are linearly independent tangent vectors of type $(1,0)$, thenAfter simultaneous diagonalization by congruence, with positive diagonal entries $a_j,b_j$, the left side iswhich is positive because some $2\times 2$ minor is nonzero.