This problem is known as bootstrapping. It's even more of a problem when designing low-voltage boost converters. If you've got a .1V 10A source, sure, you've got 1W of power, and could probably generate close to 200mA at 5V from it, but you need a voltage usable for some electronics. A 5V power supply is conveniently and indefinitely available on your devices' output soon after you solve this problem.
I won't go into design of bootstrapping a boost converter here (because I don't know how...) but I would suggest how you might go about designing a buck converter.
We'll assume that you have a circuit which can generate 12V from 120V when powered with 12V. That's not so hard, there are several designs which could do this. Wikipedia has a simple one, you might look into others in various application notes. If your switch oscillates at a few hundred kilohertz, you should be generating a usable 12V signal in a few milliseconds.
What you need, therefore, is a way to generate a voltage to power your device for just a few milliseconds, and then turn it off. A simple resistor/>12V zener diode system with a transistor just after the resistor will be fine. A PMOS with the gate to your circuit's output should switch this source off soon after your regulator starts working. You'll want to ensure that your load is disconnected when starting up, because this could cause your shunt regulator to go out of regulation.
Break your problem down into two steps:
- Generate 12V from 200V while powered from an external 12V source.
- Generate 12V from 200V for a few milliseconds without an external source.
Then combine the two. The first one is arguably more interesting, many textbooks will skip over the second problem. My professor mentioned it as a side note when lecturing.
Other answers have pointed out that you can probably just tap an AC switch-mode regulator after the rectification circuit, but I suspect that you also want to know how it works.