All pressures are measured as a differential pressure between two things:
Absolute Pressure - Absolute pressure is the pressure of a gas, referenced to a pure vaccuum. Absolute pressure is give in units of pounds per square inch, absolute, or psia. Example: Sea-level atmospheric pressure is 14.7 psia. This means the atmosphere is exerting 14.7 pounds per square inch on every surface it is in contact with, whereas a pure vaccuum would exert no pressure.
Gauge Pressure - Gauge pressure is the pressure of a gas, referenced to the current atmospheric pressure outside the chamber where the pressurized gas is. Gauge pressure is typically reported as psi, or psig. Example: The Model 3 placard tire pressure is 42 psig. This means that the pressure inside the tire is exactly 42 psi above the pressure outside the tire (atmospheric pressure).
When you use a manual pressure gauge on your tire, you are measuring gauge pressure. Let's say we fill a tire to 42 psig at sea level. There is a fixed amount of gas inside the tire, and a fixed pressure outside the tire, and they differ by 42 psi. Now we drive to 10,000 feet elevation. The same fixed amount of gas is still inside the tire, but there is now less pressure outside the tire due to the increase in elevation. Measuring the gauge pressure now will read 47 psig, because the air pressure outside the tire has dropped by 5 psi.
TPMS measures the absolute pressure of the air inside the tire. When we filled our tire to 42 psig at sea level, the absolute pressure inside the tire was 42 + 14.7 = 56.7 psia. The TPMS module reads this pressure, subtracts sea-level absolute pressure from it (56.7 psia - 14.7 psia = 42 psig) and reports that value to the car. The TPMS readings are 42 psig, which match the manual gauge at sea level.
When we drive to 10,000 feet elevation, the same absolute pressure is inside the tire (56.7 psia). The same calculation is performed because the TPMS modules do not know our elevation, nor do they know the atmospheric pressure outside the tire. They report the same 42 psig to the car.
But for the gauge, we have 56.7 psia inside the tire, and now 9.7 psia outside the tire (the elevation caused the outside atmospheric pressure to drop from 14.7 psia to 9.7 psia). The gauge measures the difference = 56.7 psia - 9.7 psia = 47 psig. This is different from the TPMS reading by 5 psi, the same as the difference in atmospheric pressure.
Now, which reading is "correct"? The answer is the gauge reading. The manufacturer of the tire guarantees it's performance based on actual forces on the rubber, steel belts, and beads. The forces are based on the differential pressure between the inside and outside of the tire, not the absolute pressure inside the tire. Thus, the gauge pressure is the one that needs to be set correctly.
As an example, if I set the tire at 42 psig at sea level, and then drive to 10,000 feet, the gauge reading becomes 47 psig. The ride becomes harsh, and the handling capability goes down. Letting air out so that the gauge pressure reads 42 psig again restores the comfortable ride.
Also note that all of the above discussion assumes constant temperature. Temperature is another variable in the tire pressure equation. Every 10 degree Fahrenheit drop in temperature lowers tire gauge pressure by 1 psig. This is because the gas inside the tire is a fixed volume, but when colder it exerts less pressure. Note that driving to a high elevation frequently causes the change in gauge pressure due to elevation and the change in gauge pressure due to temperature to offset each other. If I'm at 70F at sea level and drive to 40F at 6000 feet, there will be no change in gauge pressure. (+3 psig on the gauge from 6000 feet elevation, -3 psig on the gauge from temperature drop of 30F = no change). TPMS, however, gets affected by temperature, so TPMS gets -3 psig reading from the 30F drop and again mismatches the gauge pressure.
Examples:
Sea Level, 70F: Tire Absolute = 56.7 psia, Atmosphere Absolute = 14.7 psia, Gauge = 42 psig, TPMS = 42 psi.
6000 feet, 70F: Tire Absolute = 56.7 psia (no temperature change), Atmosphere Absolute = 11.7 psia (-3 psia due to elevation), Gauge = 45 psig, TPMS = 42 psi, Ride = Got harsher, tire overinflated.
Sea Level, 70F: Tire Absolute = 56.7 psia, Atmosphere Absolute = 14.7 psia, Gauge = 42 psig, TPMS = 42 psi.
Sea Level, 40F: Tire Absolute = 53.7 psia (-3 psia due to temperature drop), Atmosphere Absolute = 14.7 psia (no change in elevation), Gauge = 39 psig, TPMS = 39 psi, Ride = Got softer, tire underinflated.
Sea Level, 70F: Tire Absolute = 56.7 psia, Atmosphere Absolute = 14.7 psia, Gauge = 42 psig, TPMS = 42 psi.
6000 feet, 40F: Tire Absolute = 53.7 psia (-3 psi due to temperature drop), Atmosphere Absolute = 11.7 psia (-3 psia due to elevation), Gauge = 42 psig, TPMS = 39 psi, Ride = Same as sea level, correct inflation.
