Honda eu2000i 12v outlet for fan?

The generator produces unregulated AC which is rectified to DC, this circuit supplies the 12V outlet. It is then inverted to 120 VAC 60Hz. If power is bled off the DC circuit a low voltage signal will go to the inverter producing the wrong output.

This explanation is not correct. The DC output is driven from its own winding on the eu2000i's stator and has no electrical interaction with the windings that drive the 120 volt inverter. The DC output is not only unregulated, it is unfiltered. Here are some DC output measurements from my eu2000i:

With no load, output peaks are 20 volts at low idle (eco on), and 30 volts at high idle (eco off). At low idle both filaments of an 1157 bulb results in an RMS voltage of 9.2, peak voltage of 17, and a load of 1.9 amps. At high idle (recommended for charging) with the same load the RMS voltage is 14.2, peak voltage 27, and the load is 2.45 amps. On a scope, the waveform looks like ~50% duty cycle pulsating DC.

What does this have to do with a fan? I'd guess that a straight DC fan has a chance of doing ok. Most fans used in electronic devices use brushless motors and have internal electronics to create a rotating magnetic field in the stator. I don't have a clue how they'd react to the DC charging output. You could always add your own filter circuit to eliminate any potential problems.