So why does this discontinuous phase matter?
Normally it doesn't but one scenario comes mind, I'm sure there are others as well.
Consider making a Bode Plot of a Closed Loop System to retrieve the Open Loop Response. If the system under study is active and has resonances, or highly non-linear behavior, then this discontinuous nature could induce the resonance or invoke the non-linearities, all of which may have very long time constants to recovery of "Steady State". When the measurements are acquired during the Bode Plot if these effects are not allowed to die down to noise levels, then corrupted measurements are likely. Since the Bode function is generally not a true frequency "Sweep" but "Stepped" either linearly or logarithmically in frequency, this Phase Discontinuous behavior would occur at even step along the way potentially corrupting the results.
If one envisions doing Bode Plots utilizing a given AWG, then it may be wise to investigate wether the Phase Continious Mode/Feature/Behavor is available.
We found this discontinuous behavior "unsettling" at first (pun intended), but thanks to 2N3055 we are now less concerned!!
Best,