What is the meaning of system time for a multi-particle system?

When Erwin Schrödinger formulated his wavefunction equation, he was modeling a single particle that is presumed to have a position, (x,y,z) at every instant of time, t.

I don't understand what it means to apply Schrödinger's wavefunction formulation to a multi-particle system in which an ensemble of particles each have their respective positions at some given time, t. What is "system time" for a distributed system?

As I understand Relativity, one of the first things we learn is that there is no such thing as "system time" for a distributed system. Events separated in space cannot be said to be occurring simultaneously, because the notion of simultaneity breaks down in Einstein's model. Each particle in a distributed system is aging at its own idiosyncratic rate, according to its own clock. Modern atomic clocks can now measure the difference in timekeeping from moving a precision atomic clock just a few centimeters up or down in a gravitational gradient.

If there is no such thing as a common system time that pervades the universe, what does it mean to adapt Schrödinger's wavefunction to reckon a distributed system? What is John Bell supposed to plug in for time when computing his inequality?

Suppose Alain Aspect conducts his experiment at dawn on the day of the new moon — the day of spring tides. Suppose his apparatus is aligned east-west, so that one photon goes east, in the direction of the sun and the moon, while the twin photon heads west, in the direction away from the sun and the moon. There is a measurable gravitational gradient along their axis sufficient to produce spring tides on the planet. Do we still want to say that time ticks equally for the two photons? Suppose we let the two photons continue in their respective directions for 8 minutes, so that when one photon has reached the orbit of Mercury, the other photon is somewhere out near the Asteroid Belt. Now what? Does anybody really know what time it is?