Is there a way to add two complex vectors and get both magnitude and phase?

Question

Let us consider a spherical coordinate system and an electric field that has two complex components, $E_\theta$ and $E_\phi$. There is no dependence with $r$. Therefore, at any pair of coordinates $(\theta,\phi)$ you have two components $E_\theta(\theta,\phi)\hat{\theta}$ and $E_\phi(\theta,\phi)\hat{\phi}$. My basic knowledge tells me that with conventional methods I can only find the magnitude of the addition of the two by calculating as $|E_{total}(\theta,\phi)|=\sqrt{|E_\theta(\theta,\phi)|^2 + |E_\phi(\theta,\phi)|^2}$. However there is no clue or way to calculate the phase. Is there some mathematical technique to do that? Thank you in advance.