This is just a bit too long for a comment :)
Let $P$ be your polynomial and $x$ its real root. Obviously, $P(x)=0$ if and only if $$Q(x)\equiv (\mathrm{Re} P(x))^2+(\mathrm{Im} P(x))^2=0.$$ Now, $Q(x)$ is a polynomial with real coefficients, which reduces your question to finding criteria for a real-coefficients polynomial to have a real root, and these are discussed here.