Hahaha thanks
Indeed I also thought it needed to be done digitally, but even then, if you naively think about digitally shifting a 10 GHz waveform by 1 degree, it would imply that you need 3.6 TSa/s sampling rate!
I spoke to Yu Gao, a tech rep from Anapico about this, and he explained that the trick is not to shift the signal directly in time, but to just change the amplitudes of your digitised waveform, so that you get the shifted signals. With ~10 bits of resolution in the amplitude, then it is more straightforward to get the tiny phase resolution.

Thanks Steve! That was very useful.
I have a follow-up question that you might also be able to help me with. I've been looking at frequency synthesizers and noticed that some of them have phase control with >1 degree resolution. How is this done? I understand that these synthesizers use PLL to get their phase reference, but how does one then change the relative phase at the output?