Never used nor saw an Altair so I might be completely wrong in here. However this doc for some Altair 8800 contains the circuits (at the end)...
The power supply is divided to 2 zones. one with transformers and rectifiers and second with voltage regulators.
My EE skills are telling me the start race of -16V wins due to input RC filtration filter has smallest T
time to charge between all the voltages. That means either R (the first in series) is smaller or C is smaller or current is bigger...
The shut down time is related to the last R after the zener diode and or filtration caps and the device load itself. The smaller the discharge R is the quicker it shuts down ...
So by carefully selecting the input RC and output discharge resistor values among all voltages with respect to the Voltage and Current you can "program" the order of power on and shut down independently.
Here the circuit:

the input is without resistor (as for high current it would be big heat source and dissipated a lot of power) so the Voltage and Current of the transformer and sum of filtration Capacitors determine the order of power on.
This is the reason why you should recap power supply capacitors before first powering of old computer after long therm storage ... as old Caps have lost their original capacitance and the order of Voltages could be much different then designed.