The phase is a fact about where the Sun, Earth and Moon are, so it is identical in Tokyo and Lima at the same instant. Only the angle it hangs at changes.

What differs from place to place is orientation and timing: a crescent that stands up like a bowl at the equator lies on its side in Scotland, and one hemisphere sees it tipped the other way entirely. The lit fraction is the same everywhere to within a fraction of a percent, which is why a phase can be printed in a calendar for the whole world.