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Thanks guys. I do mix the fuel and oil in the tank.

So I emptied the water separator bowl completely. (I didn't replace the filter yet because it was a Sunday afternoon and the stores aren't open.) And just because I was curious, I then started the engine to see what would happen. Sure enough, within 10-15 seconds, the separator completely filled with the same greenish liquid.

It doesn't sound like that's normal, does it?