Let $1<k<d$ be an integer. Let $A \in \text{End}(\bigwedge^k \mathbb{R}^d)$, and suppose that $A=\bigwedge^k B$ for some **complex** $B \in \text{End}(\mathbb{C}^d)$.



> 
> Does there exist $M \in \text{End}(\mathbb{R}^d)$ such that $A=\bigwedge^k M$?
> 
> 
> 


In other words: Does every **real image** of the **complex** exterior map 
$$
\psi:\text{Hom}(V,V) \to \text{Hom}(\bigwedge^kV,\bigwedge^kV) \, \, \, \, , \, \, \,\psi(A)=\bigwedge^k A
$$
has a **real** source?


*Comment:* A positive answer to this question would settle [this question](https://mathoverflow.net/questions/302071/existence-of-relations-characterizing-the-k-minors-of-a-matrix).



If $A \in \text{End}(\bigwedge^k \mathbb{R}^d)$ equals $\bigwedge^k B$ for some complex matrix $B$, does it have a real source?
Counterexample: $d=3$, $k=2$, $A = -I$, $B = iI$. There is no real $M$
because $\det \bigl(\bigwedge^2 M \bigr) = \det^2 M$ and $\det(A) = -1$.


