Can you work out real power by taking the average of several measurements of instantaneous power?
Yes, although watch for aliasing.  After all, real power is the average power, usually averaged over a whole repeating cycle.

What you have to watch out for are high frequencies in the instantaneous power signal.  As when point-sampling any signal, the frequencies above half the sample rate become aliases, which is a form of noise on the sampled signal.  If the power signal has significant content above half your sample frequency, then you will get erroneous readings.

For example, consider a line-powered device that has a full wave bridge feeding a reservoir cap immediately on the power input.  The current will have significant high frequency content so that you have to sample much faster than twice per power line cycle to get meaningful readings.  Or, you can think of this same thing in the time domain.  The current will be short spikes near the voltage peaks.  If you sample such that you often miss these peaks, you are going to get the wrong idea of the average power.

Fortunately, modern microcontrollers can sample much much faster than the AC line frequency.  1000 samples per AC line cycles isn't overkill, but that's only 60 kHz sample rate, which is pretty slow compared to what many micros can do nowadays.  When trying to compute power and you think the waveforms won't be nice sinusoids, sample as fast as you can.

I should also point out that low pass filtering the voltage and current independently to avoid aliases, then computing power from these slower samples is not valid.  You can only average (low pass filter) after computing power from voltage and current, not before.  Put another way, the product of average current and average voltage does not yield average power.