Basically.....BASH is a class AB or C-ish output stage, combined with a pulsed power supply, where the rail voltages dynamically shift in level, in conjunction with the signal.Class H appears to be exactly that - class AB output with modulated power supply to minimize power losses. This scheme is used in my Benchmark AHB2 where modulating of rail voltage is part of AAA (Achromatic Audio Amplifier) patent owned by George Lucas. The key of AAA amplifier is not only modulation of the rail voltage, but also elimination of looped (recursive) feedback by adding second parallel "error" amplifier that corrects the output by subtracting the "error" form the output value (two sets of output transistors), completely eliminating TIM distortions.
Perhaps in BASH and class H power supply voltage is modulated with different time constants, but I don't see any sonic benefits of either scheme. It is used only to minimize power losses. I believe that BASH was intended for subwoofers.