A well known example of a strict-sense stationary random process is along the lines of $X_t = \sin(2\cdot \pi\cdot f\cdot t + \theta)$ where $\theta$ is some random variable, usually $\theta\sim \text{Uniform}(0,2\pi)$. f is just a fixed frequency > 0. I am slightly confused about how a realization of this process looks like. So we have
$X_t(\omega) = \sin(2\cdot \pi\cdot f\cdot t + \theta(\omega))$ and it all comes down how exactly such an $\omega$ looks like. In this example I assume $\omega$ is a scalar and is "drawn" before the start of the random process such that $\theta(\omega)$ is constant with respect to $t$. The randomness then stems from not knowing beforehand which $\omega$ would be drawn. However, after the random process has "started" $\theta(\omega)$ is just a constant phase. Do I understand this correctly? An alternative would be to draw to have $\omega = \omega_t$, i.e. a new $\omega$ for every $t$.
edit: I added the meaning of f.
