Once fixed, according to the information given in the linked technical manual, the Sony C37fet should be easily rewired/modded for standard P48 Phantom Power supply.
What’s needed:
A pair of resistors 9100 Ohm 1/4W, value-matched with a tolerance of 1% or better.
A Zener diode 10V 1/2W for protection.
This fits inside the XLR plug.
The original, 5-wire cable should be terminated to an normal 3-pin XLR male plug like this:
XLR Pin 1 and XLR case: Screen and white wire, Zener diode unmarked side.
XLR Pin 2: Red wire and resistor 1 one side.
XLR Pin 3: Black wire and resistor 2 one side.
Green wire to junction of: resistors 1+2 other side and Zener diode black ring side.
Check the voltage at the green wire to screen / pin 1, it should be ca. 8-10 V.
If you see just 0.7 V the Zener diode is reversed.
Explanation:
C37fet is already terminated for use with an external power supply.
The factory original cable has two extra wires for this purpose:
White for the negative PSU polarity.
Green for the positive PSU polarity.
The mic is drawing a current of 5 mA, which is in the range the P48 Phantom Power can supply.
So it’s just about dimensioning the series resistance to achieve the correct 9 V supply voltage.
P48 is made to be exactly equally loaded from both hot and cold signal wires, therefore two resistors matched for the same value are used to feed the mic.
Absolute resistor value is not important.
The C37fet is designed to work from 7-10 V battery voltage.
The 10V Zener diode is for over-voltage protection in case the mic draws less than the specified 5 mA of current, or in case the standard P48 supply works out of standard, to prevent the capsule bias going too high.