I've come across this high voltage 'modulator' circuit and the MOSFET gate drive circuit is shown below. The floating wire on the bottom connects to a duplicate of this circuit at the MOSFET drain (whose source is connected to -V1) with the 'dot' flipped on the duplicate's secondary:
(Don't pay too much attention to the component values -- especially M1 and L1. There's also a TVS on the MOSFET gate not shown here.)
Here L1 is the secondary winding of the 1:1 isolation transformer. The primary side pulses +15V at about 1% duty, then -15V at the same duty, and repeat. This charges the gate of one MOSFET at a time, while simultaneously draining the gate of the other (the other winding will see a negative pulse). I figured out that C1 and D3 (12V Zener) hold the gate open after the positive pulses, and closed for the negative.
But what is the purpose of the anti-parallel switching diodes, D1-D2? They appear to be in a clipping/limiting configuration, but I cannot figure out why they're needed.