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?

I think the easiest example to come up with is $e^{2i\pi/3}$,
$$e^{2i\pi/3}\cdot e^{2i\pi/3}\cdot e^{2i\pi/3} = e^{2i\pi}=1.$$
