I'm searching for rather specific counter-example.

Some notation: $A(\alpha|\beta)$ is the sub matrix of $A$ with with rows $\alpha$ and columns $\beta$. $\textrm{det } A(\alpha|\alpha) =: \textrm{det } A(\alpha)$ are the principal minors of $A$.

Is there a invertible, irreducible matrix $A \in \mathbb{C}^{4 \times 4}$ with $\textrm{det } A(\alpha) = \textrm{det } \overline{A}^{-1}(\alpha)$ for all $\alpha$ and $\textrm{rank } A([1,2],[3,4]) = 1$?