I would be inclined to replace all of the tantalum capacitors and both ‘L’ voltage regulators.
With the components removed (and the desoldering tidied up) check there are no short circuits between the incoming +/- 15V rails and 0V/GND. If this is correct, power up and make sure your ‘raw’ supplies are Ok at this stage.
Also, check there are no short circuits from the output of each of the ‘L’ voltage regulators and 0V/GND. You might want to consider doing this before powering up (above).
Solder the ‘input’ tantalum bead capacitors in place and test again.
I would now remove the 1488 and 1489 RS232 drivers next and install the two ‘L’ voltage regulators and test again. You should now observe stabilised +/- 12V on the output of the regulators.
Solder the ‘output’ tantalum bead capacitors in place and test again.
Install the 1488 and 1489 and test again.
Basically, proceed slowly and in an ordered manner.
Good luck...
Dave