Am I violating any causal inference assumptions? Consider a large-scale AB test with two variants: control and treatment. Assume we want to compare the two variants with respect to the following metric: "Proportion of users who streams more than X minutes"
Now to the controversial part: Assume we set X to the median streaming time of Control users.
Are we then violating any assumptions? Are there any concrete examples where this can have meaningful implications on the result?
 A: The research question is statistically isomorphic to asking whether the two groups have the same median. Let's say the proportion of units in both groups who stream more than the median streaming time in the control group is the same. In the control group, this is 50%, by definition of the median. In the treated group, this must also be 50%, assuming the proportions are the same. The point in the treated group where 50% of people watch more is the median of the treated group. Therefore, the median streaming time in the control group is equal to the median streaming time in the treated group. Framing your hypothesis in this way is far less controversial and is amenable to statistical tests (e.g., Mood's median test). 
I don't see a reason why the proportion in the treated group greater than the median in the control group can't be used as a descriptive statistic to supplement a test of the difference in medians. It could even be used as a test statistic, though the test would probably have to be a bootstrap or permutation-type test that accounted for the fact that the estimated median in the control group would differ among bootstrap samples or permutations. 
It would be important to note that your research question would have to be framed in such a way as to avoid $X$ being considered fixed since it isn't; you would have to make it clear that your research question concerns the statistical quantity "the median of the control group in the population", which is estimated in your sample but not observed. If you wanted to make your research question about "the observed value of the median in this single control group sample", you wouldn't have to take into account the variability in the median, but the comparison wouldn't tell you anything about how the treated group "population" compares to the control group "population", which is really what you want.
