What Information does the phase of a cross power spectra give me?
If the two signal show a significant correlation at some frequency $f$ then the phase measured in the cross spectrum gives you the phase lag of the signals at that frequency.

So, take $$ A = \sin(2\pi f t ) $$ and $$ B = \sin(2\pi f t - \phi) $$ then the cross spectral phase at frequency $f$ will give you $\phi$.