I don't know either and await,with interest,the responses from persons who do. Yes,I have muddled through the posts about "time coherence",much of which is beyond me.
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They basically do three things: reduce the "beaming effect" of high frequencies, reduce the reflection of sound waves that can gather towards the center of the driver, and allow more air into the coil. The first two can muddle the driver's overall frequency response and the third can limit the power handling of the driver. Some of the better drivers employ phase plugs (Thiel, Vifa, Lowther, Scan-Speak) and there does not seem to be a trade-off that results (other than cost).
The way they perform their "purpose" is by preventing the frequencies from interfering with each other at the center of the driver - much in the same way a propeller cone prevents water (or air) from cavitating at the center, and by eliminating the need for a dust cap at the center which acts as a springboard for higher frequencies to beam from. And by physically being through the center of the driver, they allow air to enter the cone from front and back, improving ventilation.
One thing to note is that for a phase plug to work, it has to be attached (fixed) to the coil/magnet and not float on the cone itself.