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5$\begingroup$ Once more, Kerbal Space Program provides a useful example. (Duna, KSP's Mars-analogue, has a single huge moon called Ike that it's mutually tide-locked with, somewhat like the real-world Pluto-Charon system.) $\endgroup$– Ilmari KaronenCommented Feb 20, 2017 at 9:13
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3$\begingroup$ Yes, the moon would wax and wane in one day. The phase has little to do with the observers location, it is purely how much of the moon that is facing the sun that the observer can see. Two observers at different longitudes would see different phases. $\endgroup$– Paul SmithCommented Feb 20, 2017 at 12:05
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$\begingroup$ @PaulSmith, wait. So different latitudes see different moon positions and different longitudes will see different phases at the same time for the same moon position???. Observers would also see the moon either a little bigger or smaller depending on their longitude??? $\endgroup$– EveryBitHelpsCommented Feb 20, 2017 at 12:44
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1$\begingroup$ @EveryBitHelps Yes. Text moved to an answer. $\endgroup$– Paul SmithCommented Feb 20, 2017 at 13:43
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1$\begingroup$ @EveryBitHelps Yes, and those effects will be significant (in astronomic scale). For example: geostationary orbit has radius of ~35700 kilometres, while Earth radius is ~6400 kilometres. Observer at equator, where moon is in zenith is 29300 km away, while observer at pole or at any longitude 90 degrees away from moon-zenith spot would be 36300 km away from moon, and there moon will appear 35% smaller. $\endgroup$– M i echCommented Feb 20, 2017 at 17:30
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