Is it true that the average of the product of current and voltage is always real power?
If you average the product of the current and voltage in this
  situation, you’d get zero.


Think again...



I think you need to look at the top left diagram (resistive load).

Picture from this answer. See also these answers: -


What's the most economical way to digitally measure 240V mains voltage, current and power factor?
Average Power Versus Real Power
Measuring AC power usage



  If you have an AC voltage waveform centered around gnd over some
  complex load that causes distortion/deformation in the current signal
  so that the current is not purely sinusoidal, how would you measure
  the power?


Quite simply instantaneous v multiplied by instantaneous i then averaged will always give you real power consumed (i.e. what you would be billed on by your utility company).


  What if the AC voltage waveform was slightly offset?


Exactly the same answer: Average(v x i) is power.