Appreciate this discussion! Phase coherent (drivers matching in time at the crossover frequency) is not the same as flat phase (literally a flat line when measuring phase response) Most speakers when measured will show a phase "wrap" which is a time domain behavior error and is separate from drivers being "in time" at their crossover frequencies.
Just looked up the Kii speakers and I didnt see anything about the level of dsp correction and will assume it is the basic level of eq/crossover/time delay and limiting.
This speaker uses the David Gunness temporal processing
https://www.presonus.com/products/Sceptre-S6
This is an example of a JBL speaker with V5 tuning. Note the author uses words like "incredibly detailed, smooth, etc) not words you normally associate with PA speakers. I can tell you the even EAW's least expensive powered speakers with FIR "focusing" have a very wide deep soundstage, reverb tails that go forever, etc, again, not how you would normally describe PA speakers.
https://www.svconline.com/industry/strother-bullins-reviews-jbl-srx835p-402893
Here is some more detail from David's current company. He and John Meyer were really the pioneers on flat-phase processing (to my knowledge)
https://www.fulcrum-acoustic.com/audio-technology-insights-resources/temporal-equalization-tq/
Bluehorn from Meyer
https://meyersound.com/news/first-bluehorn-patent/ "The system-specific correction algorithms hosted by the processor were developed by meticulously analyzing all physical and electrical non-linearities throughout the entire system, from 25 Hz to 20 kHz, in order to bring acoustic output – at any level – into phase and frequency alignment with the input signal. Essentially, the digital processing nullifies the non-linearities of drivers and enclosures, resulting in a phase-coherent response previously achieved only by open-air electrostatic headphones"
It's difficult to describe the difference in words other than the sound is "put together" in a natural sound way. Being able to A/B the same speakers with standard processing and time-based FIR processing is eye-opening. When mixing live sound, the instruments dont "fight" each other in the mix but blend more naturally like how they blend in real life. Its difficult to describe but hopefully my explanation makes some sense?