I think that there must be something wrong with your equations. The first equation implies that $\psi_x\phi_y-\psi_y\phi_x=0$, but this iimplies that $\phi$ and $\psi$ cannot be independent functions of $(x,y)$. Surely you have a typo somewhere.

OK. I didn't know whether you wanted to deal wth the unoriented case or not, so I wasn't sure that my comment was relevant. I'll write my comment as an answer and put in a little more detail tomorrow, when I have access to Lawlor's paper.

It seems that you are asking whether a pair of orthogonal 2-planes in 4-space is area-minimizing. This is true in the space of integral currents (no matter how the plane are oriented), and I believe it's also true when you consider non-orientable surfaces. The general criterion for a pair of oriented $p$-planes in $2p$-space to be area-minimizing among orientable $p$-dimensional surfaces is known as the Angle Theorem (G. Lawlor, The Angle Criterion, Invent. Math. 95 (1989), 437–446.)

@MikhailKatz: Neither condition ($\mathrm{d}\omega=0$ vs. $J$ integrable) implies the other when $n>1$. The two together are the definition of Kähler.

@MikhailKatz: I think that one way of describing what you are asking is whether, for a symplectic $2n$-manifold $(M^{2n},\omega)$ (i.e. $\mathrm{d}\omega=0$ with $\omega^n$ nowhere vanishing), every `$\omega$-compatible' metric $g$ (i.e., the endomorphism $J$ such that $g(v,w) = \omega(v,Jw)$ satisfies $J^2=-I$), has its associated $J$ be integrable. The answer to this is no for $n>1$. (Of course, the answer is yes for $n=1$.)

@BenJohnsrude: The question is about 2-forms, not 1-forms, so the triple wedge is actually a symmetric product, not an alternating one.

Using the lower bound in the paper joro cites in his answer to mathoverflow.net/q/230740, it follows that there is a positive constant $c$ such that there are infinitely many positive integers $m$ such that $xy(x+y)=m$ has at least $c(\log m)^{1/2}$ integer solutions $(x,y)$. Since $m$ has to be even (else there are no integer solutions $(x,y)$), we only have to make sure that are enough solutions with $x$ and $y$ positive. But since $(x,y)$ is a solution if and only if $(x, -x-y)$ and $(-x-y,y)$ are solutions, there are at least $c/3(\log m)^{1/2}$ solutions $(x,y)$ with $x,y>0$.

@user1234567890: In the case that $\det G$ vanishes at an isolated point, this can be reduced to a topological question, because then the $(p,q)$-type of $G$ away from the isolated singularity must be constant, and you are essentially asking when a map from a punctured $m$-disk (which is homotopic to $S^{m-1}$) into the homogeneous space $\mathrm{GL}(n,\mathbb{R})/\mathrm{O}(p,q)$ can be lifted continuously to $\mathrm{GL}(n,\mathbb{R})$.