If two real matrices are conjugated over $\mathbb{C}$, are they then also conjugated over $\mathbb{R}$? As in the title: If two real (square) matrices are conjugated over $\mathbb{C}$, are they then also conjugated over $\mathbb{R}$?

The answer to this question is yes. For one, it suffices to note that any real matrix is similar to some matrix in real Jordan form, and that any two similar real matrices share such a real Jordan form.

There may be a way to show this directly that doesn't use such powerful techniques, but this certainly works.