I don't know of a real example, but maybe I can provide some helpful thoughts nonetheless.
The first thing is that the nature of "Simpson's paradox" has evolved over time. Today, it is widely known as the situation where there is a relationship between two variables (call them $X$ and $Y$) with a given direction, but when including information about a grouping variable ($Z$) that was not previously included, the direction of the relationship between the two variables flips. This is a specific case of a general phenomenon in which relationships can change or even reverse when including more information. It is due to the fact that the two covariates, $X$ and $Z$, are correlated. In general, today it is typically understood that Simpson's paradox refers to a situation with observational data and where the relationship between $X$ and $Y$ controlling for $Z$ is the 'true' one.
The paradoxical effect of the sign flipping was not the point of Simpson's (1951) paper, however. That this could occur was known much earlier (Yule, 1903). For example, Simpson wrote, "The dangers of amalgamating 2 x 2 tables are well known..." (p. 240). Instead, Simpson's point was that you can't say a-priori that either the disaggregated or aggregated analysis will provide the 'right' answer. You have to know the question, and depending on that, either could be correct. It may be helpful to quote his examples:
An investigator wishes to examine whether in a pack of cards the proportion of court cards (King, Queen, Knave) was associated with colour. It happened that the pack which he examined is one with which Baby had been playing, and some of the cards were dirty. He included the classification "dirty" within his scheme, in case it was relevant, and obtained the following probabilities:
Court Plain Court Plain
Red . . . 4/52 8/52 2/52 12/52
Black . . . 3/52 5/52 3/52 15/52
It will be observed that Baby preferred red cards to black and court cards to plain, but showed no second order interaction on Bartlett's definition. The investigator induced a positive association between redness and plainness both among the dirty cards and among the clean, yet it is the combined table
Red . . . 6/52 20/52
Black . . . 6/52 20/52
which provides what we would call the sensible answer, namely that there is no such association.
Suppose we change the names of the classes in Table 2 thus:
Untreated Treated Untreated Treated
Alive . . . 4/52 8/52 2/52 12/52
Dead . . . 3/52 5/52 3/52 15/52
The probabilities are exactly the same as in Table 2, and there is again the same degree of positive association in each of the 2 x 2 tables. This time we say there is a positive association between treatment and survival among both males and females; but if we combine the tables we again find there is no association between treatment and survival in the combined population. What is the "sensible" interpretation here? The treatment can hardly be rejected as valueless to the race when it is beneficial when it is applied to both males and females.
So the point here is different than what Simpson's paradox has become. It is more subtle, and in my opinion, more interesting. What is the 'right' way to analyze a dataset depends on what you are trying to accomplish.
In my opinion, the DAG from Pearl that you quote doesn't match what people typically understand as 'Simpson's paradox'. That is, it isn't a case of observational data that are confounded. Instead, the treatment ($X$) seems to be an exogenous cause. In that case, controlling for blood pressure ($Z$) is conditioning on a (partial) mediator. If you did that, it would weaken the total effect measured, because you would only assess the $X \rightarrow Y$ path, whereas the total effect is the sum of both the $X \rightarrow Y\; \&\; X \rightarrow Z \rightarrow Y$. When you lessen the effect measured, it may even become non-significant, depending on the power of the analysis. I'm not saying that Pearl is wrong or that the example is useless. I'm arguing that we need to be very clear and explicit regarding what we're talking about and what we are supposing the investigator wants to achieve.
Simpson's counterexample, quoted above, is observational / descriptive in nature. We can also consider a predictive context. With predictive modeling (cf., Shmueli, 2010) the goal is to be able to use the developed model in the future to predict unknown values. It doesn't matter if you have the 'right' $X$ variables, and the relationship between $X$ and $Y$ is not of interest. What matters is whether a predicted value matches the true value with sufficient accuracy. In the typical examples of Simpson's paradox, the confounding grouping, $Z$, is usually implied to be obscure. Now, imagine a predictive situation in which I can get more accurate predictions by taking $Z$ into account, but the model would perform worse if I didn't have the $Z$ values, and end users are extremely unlikely to have them. In that case, a predictive model built without $Z$ would be unambiguously better.
Again, that example (such as it is) reflects a different situation with different goals. If you want something that sounds like Pearl's example, consider this: One of the things the doctors who manage emergency rooms are most interested in, is how to move patients through more quickly. There are a couple things to bear in mind here. First, there are generally three paths that patients follow: 1) discharged to home, 2) admitted to hospital, and in between, 3) held for observation for a period of time and then either discharged or admitted. The lengths of time involved is 2 > 3 > 1, with near perfect separation between the three paths. The second thing is that doctors, especially in the ER, are risk-averse. In ambiguous situations, they defer to more extensive treatment, which in this case means a slower path through the ER. Now, imagine a new protocol (checklists, additional tests, etc.) is developed for patients presenting with a certain condition. Implementing this new protocol, on top of everything else that's done, makes each path take longer. However, it yields more appropriate treatment and, importantly, clarifies much of the ambiguity that would have otherwise existed. That means many patients will move through a shorter path than they would otherwise. In this example, an exogenous intervention / treatment ($X$) makes the time through the ER slower within each path / group ($Z$), but isn't independent of group. Moreover, group membership has a large effect on time ($Y$). But the "sensible" interpretation is the change in the marginal distribution of $Y$.
- Shmueli, G. (2010). "To Explain or To Predict?", Statistical Science, 25, 3, pp. 289-310, 2010.
- Simpson, E.H. (1951). "The Interpretation of Interaction in Contingency Tables". Journal of the Royal Statistical Society, Series B. 13, pp. 238–241.
- Yule, G.U. (1903). "Notes on the Theory of Association of Attributes in Statistics". Biometrika, 2, 2, pp. 121–134.