If $\{X_t\}$ is the process defined by $$X_t=\sum_{j=1}^n A(\lambda_j)e^{it\lambda_j}$$ in which $-\pi<\lambda_1<\lambda_2<...<\lambda_n=\pi$, and $A(\lambda_1), ..., A(\lambda_n)$ are uncorrelated complex-valued random coefficients (possibly zero) with zero mean. Then $\{X_t\}$ is real-valued if and only if $\lambda_j=-\lambda_{n-j}$ and $A(\lambda_j)=\overline{A(\lambda_{n-j})}$, $j=1,...,n-1$ and $A(\lambda_n)$ is real.
This is problem 4.4 from Brockwell & Davis. The "if" direction is easy, but the "only if" seems very difficult. I can only prove the almost trivial $n=1,2$ cases. Any suggestions?