Extract same in phase signal from 3 different sources with shared delayed noise?
Depending on how good your approximation of x(t) is, you might want to try this:


Subtract your approx. of x(t) from all $S$
Cross correlate the signals to get $\phi_2$ and $\phi_3$ (choose $\phi_1$ = 0)
You now should have n(t), 
Subtract $n(t)$ from orig $S_1$.
You now have x(t)


edit: Please note, that this will only work if your approximation of x(t) is quite good, and the noise you have really is equal in all mics (and not introduced due to electronics etc.)