Conjugate in $\mathbb{C}$ implies conjugate in $\mathbb{R}$? If $A,B\in M(n,\mathbb{R})$ and there exists $P\in GL(n,\mathbb{C})$ such that $A=PBP^{-1}$, does that imply that there exists $Q\in GL(n,\mathbb{R})$ such that $A=QBQ^{-1}$?

Yes, since they both are conjugate to the rational canonical form. (The Wikipedia calls this Frobenius canonical form.)
