electromagnetism - Unpolarized wave, $Deltavarphi =Delta varphi(t)$?

I have seen a unpolarized wave defined in a number of places (e.g. here) as a wave such that: $$\begin{align} E_x&=E_0 \cos(kz-\omega t) \\ E_y&=E_0 \cos(kz-\omega t+\varphi) \end{align}$$ Where $\varphi=\varphi(t)$ is a random function in time.
My question is why do we not have $\varphi=\varphi(x,y,z,t)$ with it been a random function in time and space?

(This question follows from discussion in the comments of: Introducing a phase, what changes?)