real world examples of causality without correlation There are several hypothetical instances of causality despite the lack of correlation between two variables, some of them have been mentioned in this forum - Does no correlation imply no causality?
Would anyone be able to share examples of real-world published cases in which there was clear causality without (linear) correlation?
 A: Yes, of course, we can have examples of real-world cases where we have a causal relationship without a strong correlation; an obvious place to check is causal time-series models. Assume for example we have the structure:

It is easy for to say that $Y_t\perp \!\!\!\perp X_{t-1}|Y_{t-1}$ based on $d$-separation, i.e. we have conditional independence between $X_{t-1}$ and $Y_t$ given $Y_{t-1}$. Here, only the value of $X_t$ would be assistive to predict $Y_t$ and thus Granger causality (or some other "basic correlation" method) would not detect the influence of $X$ on $Y$ because the past of $X$ influences $Y_t$ only via the past of $Y$. As a result, the causal relation of $X$ on $Y$ is completely missed based on correlation measurements; I "borrowed" this example from Ch. 10 in "Elements of Causal Inference" (2017) by Peters et al., check it out for more formal treatment. Now for a published result: "A study of problems encountered in Granger causality analysis from a neuroscience perspective" (2017) by Stokes and Purdon is a quick one; they use a VAR model instead of a simple ARX as above but the logic is the same - a vector auto-regressive (VAR) model generalising a single-variable autoregressive (AR) model by allowing for multivariate time series. In general, search the terms "Granger Causality" and "Common-cause fallacy" alongside each other and a few references pop up; "A review of the Granger-causality fallacy" (2015) by Maziarz is a good place to start too.
