I have commented before, but an additional comment might be in order.That is that terminology often obscures the discussion. Some of that obfuscation is purposeful, some is accidental. It may help to insert the word ’polarity’ when it fits. Polarity, here, is the direction the driver moves when fed a signal. A 6-volt dc (battery) signal is appropriate. With plus to plus, the driver(s) should move out into the room. Coherent speakers do that, most don’t. A Wilson, KEF, etc. will leap-frog such that the woofer comes out, midrange goes in, tweeter comes out, etc. Those speakers are polarity-incorrect. Conversely, when all drivers come out, many companies call that ’phase correct’, despite any other phase or timing anomalies. I would call that ’all out’ condition ’polarity correct’. But most designers and critics consider either behavior as OK, whether all drivers move out, or leapfrog up the array, irrespective of many phase and time anomalies. That’s because the ear-brain can reassemble the intended wave-form, and do it quite well. I believe that many listeners, including pros, actually enjoy the mental gymnastics required to reassemble the waveforms. Consider that the BBC ruled that if an individual person can distinguish positive or negative air pressure (polarity inversion), the preference-judgement is personal. Their ’research’ demonstrated that a majority of their subjects preferred negative polarity, which would make a drum hit (for instance) suck rather than blow. The leading edge would be a vacuum whereas it was a pressure wave from the real drum. The BBC deemed that the negative pressure attack was more polite and acceptable. - preferred by more (British) listeners.
In that light, the typical designer generally strives for ’listenability’, that polite, acceptable presentation, which is quite often not what the microphone ’heard’ or the recording stream produced. (Deeper discussion deferred that many ’modern’ recordings invert various polarities to ’fill the mix’.) But I am addressing the speaker reproducing its input signal. Wilson (as example) inverts polarity at each driver exchange (crossover point.) Good engineering executes the hand-off between drivers with smooth phase transitions. The absence of abrupt glitches gives the ear-brain no hard evidence of trouble. And most design styles, companies and critics call that victory.
In contrast Jim Thiel, Richard Vandersteen (and a few other oddballs) chose to preserve the phase-time information intact. Many of you guys appreciate that, most people do not. When phase is kept intact AND the drivers are aligned so that their leading edge transients all reach the ear simultaneously (time-alignment), we call that Coherence. I notice that today the C word usually means ’smooth phase transitions’. rather than our assignment of ’integrated waveform’ period. First order roll-offs (including electrical and acoustic elements) sum to produce zero phase shift. (One driver leads by the same amount that the other lags, such that at the design listening distance, they sum to produce no shift.) Add physically equidistant sound sources, and you get an actual representation of the input signal representing the recorded sound, with no need for the brain to descramble the phase and time information. Thiel and Vandersteen decided that goal of authenticity was worth all the difficulty of making it right. The industry at large does not consider that element of fidelity to be important, or important enough to warrant its difficulties.
I suggest pulling out ’phase’, ’polarity’ and ’time’ in trying to understand the landscape. Richard Hardesty’s journal has been cited here. I consider his clarity and teaching style to be stellar.