Since the eigenvalues of a real matrix $A(\theta)$ come in complex conjugate pairs, an eigenvalue on the real axis with multiplicity 1 cannot move off the real axis when $\theta$ is varied over a small interval. It must first merge with a second real eigenvalue. This means that the set of $\theta$ where $A(\theta)$ has at least one real eigenvalue consists of the union of intervals $[\theta_1,\theta_2]$, $[\theta_3,\theta_4]$, $\ldots$.
Generically, an $n\times n$ real matrix $A$ will have on the order of $\sqrt{n}$ real eigenvalues, so if $n$ is large and you pick a random $\theta$, you are likely to find at least one real eigenvalue. No fine tuning of $\theta$ is needed. This seems the most efficient way to solve the problem.