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Are tidal effects from the Sun and/or Moon taken into account in GPS systems?

More broadly, is the heliocentric system used somehow in GPS?

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Are tidal effects from the Sun and/or Moon taken into account in GPS systems?

On calculating the orbits of the satellites, yes. The satellites orbit high enough that accurately modeling the orbits of the satellites mandates accounting for third body perturbations from the Moon, the Sun, and the planets. The orbits of the satellites are calculated from the perspective of a geocentric frame, the relative positions of the perturbing bodies are calculated from the perspective of a solar system barycentric frame (which answers your second question).

On the timing, no. Imagine a GPS satellite in opposition (on the line between the Sun and the Earth and on the opposite side of the Earth than the Sun) and a person on the Earth who sees the Sun as being straight overhead. The effect of the Sun on the tick rates of the satellite's clock and the person's clock is very small, less than 0.2 microseconds/day. The satellite doesn't hang suspended at that position. It orbits the Earth. Half an orbit later, the satellite will be close to being in conjunction (where it's clock will tick a tiny bit slower). That already small error of 0.2 microseconds/day will average out to near nothing as the satellite orbits the Earth.

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Yes, though the effect is greater on earth-bound or terrestrial tracking stations than on spaceborne or orbiting satellites. See Table 2.9 ("Perturbing accelerations acting on a GNSS satellite") and sec. 10.1.2 ("Site Displacement Modeling: Solid Earth Tides, Pole Tides, and Permanent Tides") in the BERNESE Software Manual

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