I still don't understand in which sense do you use the word "real" in 'correlation is "real"'.
Let's say you have two time series 100 data points in length each. You calculate their correlation, say, Pearson's correlation. It's a number. In which sense can that number be "real" or "not real"?
Do you implicitly have in mind the sampling theory where what you observe is a sample estimate and what's "real" is the true parameter of the unobserved underlying process? In this case there is a very large body of research that mostly goes by the name of "frequentist statistics" about figuring out what does your sample estimate tell you about the unobserved true value (to call which "real" is a bit of stretch since normally it is not real).
It seems as though my attempts to define my term intensionally aren't working, so I'll try and give an extensional definition instead:
An example would be that site you linked earlier. Those quantities appear to be correlated, but the correlations are not "real".
Another month, another rationality quotes thread. The rules are: