Sigh.

All measured correlations are "actually present in the data". If you take two data series and calculate their correlation it would be a number. This measured (or sample) correlation is certainly real and not fake. The question is what does it represent.

You claim the ability to decide -- on a completely unclear to me basis -- that sometimes this measured correlation represents something (and then you call it "real") and sometimes it represents nothing (and then you call it "not real"). "Redundancy" is not an adequate answer because all it means is that you will re-measure your sample again and, not surprisingly, will get similar results because it's still the same data. As an example of "not real" correlation you offered the graphs from the linked page, but I see no reason for you to declare them "not real" other than because it does not look likely to you.