Wedge positivity for two positive (1, 1)-forms
= Wedge positivity for two positive (1, 1)-forms
If $\varphi$ and $\psi$ are positive real $(1,1)$-forms and $\xi,\eta$ are linearly independent tangent vectors of type $(1,0)$, then
$$
(\varphi\wedge\psi)(\xi,\eta,\bar\xi,\bar\eta)>0.
$$
After simultaneous diagonalization by congruence, with positive diagonal entries $a_j,b_j$, the left side is
$$
\sum_{j<k}(a_jb_k+a_kb_j)|\xi_j\eta_k-\xi_k\eta_j|^2,
$$
which is positive because some $2\times2$ minor is nonzero.