If you are mixing two independent sounds, the lock-in or phasing of the underlying timers should not matter. This is entirely in the audio domain.
If you are trying for volume amplification, with matching but complementary outputs, then you need the phase matching. With various harmonics are differing frequencies, and therefore at differing phase relationships, you might only get volume amplification on some frequencies, while getting cancellation on some of the harmonics. It might actually sound quite cool - I know people who play with exactly those features - but they do that precisely because it is unpredictable. "Unpredictable" is usually not a desirable feature when you're aiming for amplification.
As to the ability to keep the complementary outputs without the original - let's just say, that is one really complex section of datasheet!
10.3.3 Alternate Functions of Port D, • T1/OC.4D/ADC9 – Port D, Bit 6
OC.4D: Timer 4 Output Compare D. This pin can be used to generate a high-speed PWM signal from Timer 4 module, complementary to OC.4D (PD7) signal. The pin has to be configured as an output (DDD6 set “one”) to serve this function.
I don't think there is any doubt on this point - if the complementary signal is generated, we can route it out this pin.
On to 15.12.3 TCCR4C – Timer/Counter4 Control Register C, particularly • Bits 3,2 - COM4D1, COM4D0: Comparator D Output Mode, Bits 1 and 0.
The sentence that would give me doubt is "The complementary OC4D output is connected only in PWM modes when the COM4D1:0 bits are set to “01”."
So - when do we set 01? My reading of tables 15-15, 15-16, 15-17 is that in Normal (Non-PWM) mode (Table 15-15) we do indeed lose the complementary output. (This would seem to contradict the sentence above.) But in Table 15-16. Compare Output Mode, Fast PWM Mode and Table 15-17. Compare Output Mode, Phase and Frequency Correct PWM Mode, I note that when setting 01 (which has different meanings in the two tables), the complementary output is listed as connected, but disconnected in all of the other three modes.
Finally, the block diagram for the timer/PWM section strongly suggests that the 'connected' and 'disconnected' applies inside the block. This would seem to indicate that, as long as the signal is 'connected' in the PWM block, it is available for the alternate function of the pin.
So - it's back to the libraries. Is the 01 mode in either Fast PWM Mode or the Phase and Frequency Correct PWM Mode the setting that is used in the libraries? I would have to hand that question back to someone extremely familiar with those libraries. If the libraries use different modes, then we lose the complementary output. If they do use the 01 mode, then it appears we keep the complementary output.
I've tried to be almost excessively complete, so that if I am wrong, it's far easier for someone to point out where my logic chain broke.