How do you measure phase difference between two wideband signals
It's a hard problem.  You can FFT the signals, compute the phase and amplitude of all components, and (except for an arbitrary integer times 2pi) find a weighted average phase for each, and take the difference.   There's gonna be more than one component, though, so the 'phase difference' isn't a constant, it's a component-by-component listing.

If you consider only one component, you've filtered the signal under consideration by making that choice, and the bandwidth is different from the original signal.