Phase shifting a noisy signal I have a signal of the form $s(t)=A(t) \sum \cos(\omega_i(t)t +\phi_0) + n(t)$, where $n$ is gaussian noise.

I can only read the signal+noise and thus can not separate them.

I want to phase shift the signal to $A(t) \sum\cos(\omega_i(t)t)+n'(t)$ and I am at a loss on how to do this. During lectures / courses I've always done phase shifts simply by multiplying with $e^{i\phi_0}$.

My signal is of the form $s(t)=A(t) \sum \cos(\omega_i(t)t +\phi_0) + n(t)$, as opposed to $s(t)=A(t) \sum e^{\omega_i(t)t +\phi_0} + n(t)$, so I can not simply multiply with $e^{i\phi_0}$

Is there any way to do this? I'm asking because I am interested in cross-correlating two signals.
With a real cosine a phase shift is equivalent to a time delay.  If you delay the signal by $\phi_0$ then you will get the result you want.

The number of samples that you will need to delay by will likely not be an integer number.  You can delay by a non-integer number of samples via a fractional delay filter.