It seems to start with the tail end of the Andalusian cadence, then resolving to the tonic, before continuing on to the bIII degree.
For instance we can loop this and then finally just end on the tonic.
That could be regarded as the bVI-V pickup or anacrusis, where the tune starts on the i tonic:
(bVI-V) i-bIII-bVI-V i-bIII-bVI-V i
pickup ^ start of ^ repeat ^ end
It almost seems like a kind of variation on
i-bVII-bVI-V, which is the Andalusian Cadence1. Does it have its own name?
The bVII is replaced by bIII, which gives it a rock/blues mood, but that Phrygian ending retains the Flamenco flavor.
i-bIII-bVI-V has come up in some questions on this site and is googleable.
The idea that this is an alteration of the Andalusian Cadence, has some support here which notes the possibility of a bIII chord substitution, with the bassline unchanged.
So that is to say if the original bass line (e.g. AGFE in A minor/E Phrygian) is prominently present, it meshes with this chord progression and emphasizes that relationship to the Andalusian.
- The Andalusian Cadence is also written IV-III-II-i, with respect to a Phrygian tonic, which makes sense in context of the musical culture and practice from which its name is derived.