Can the product of three complex numbers ever be real? Say I have three numbers, $a,b,c\in\mathbb C$. I know that if $a$ were complex, for $abc$ to be real, $bc=\overline a$. Is it possible for $b,c$ to both be complex, or is it only possible for one to be, the other being a scalar?

Another approach: suppose $a, b$ are complex and not real and $ab$ isn't real. Then let $c=\overline{ab}$.
Note that in a precise sense this is universal: if $abc$ is real (and each is nonzero), then $c$ is a real multiple of $\overline{ab}$.
