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It's sort of hard to describe. Depending on which partial has had phase changes, there can be obvious timbral changes...or nothing noticeable at all. The phase-change effects get more apparent as the partials get closer together in frequency, and ultimately wind up affecting each other more than the fundamental at that point.

Of course, this is only referring to fixed phase changes. If the phase changes are modulated, then a lot of timbral evolution/alteration becomes apparent, since human hearing has problems with hearing phase in of itself, but if phase of a frequency is shifting against something else, we perceive it much more readily.