Imagine that the only way that civilization could be destroyed was by a large pandemic that occurred at the same time as a large recession, so that governments and other organisations were too weakened to address the pandemic properly.
Then if we looked at the past, as observers in a non-destroyed civilization, what would we expect to see? We could see years with no pandemics or no recessions; we could see mild pandemics, mild recessions, or combinations of the two; we could see large pandemics with no or mild recessions; or we could see large recessions with no or mild pandemics. We wouldn't see large pandemics combined with large recessions, as that would have caused us to never come into existence. These are the only things ruled out by anthropic effects.
Assume that pandemics and recessions are independent (at least, in any given year) in terms of "objective" (non-anthropic) probabilities. Then what would we see? We would see that pandemics and recessions appear to be independent when either of them are of small intensity. But as the intensity rose, they would start to become anti-correlated, with a large version of one completely precluding a large version of the other.
The effect is even clearer if we have a probabilistic relation between pandemics, recessions and extinction (something like: extinction risk proportional to product of recession size times pandemic size). Then we would see an anti-correlation rising smoothly with intensity.
Thus one way of looking for anthropic effects in humanity's past is to look for different classes of incidents that are uncorrelated at small magnitude, and anti-correlated at large magnitudes. More generally, to look for different classes of incidents where the correlation changes at different magnitudes - without any obvious reasons. Than might be the signature of an anthropic disaster we missed - or rather, that missed us.