Meaning of "a statistic $U$ is ancillary to another statistic $T$"?

Question

From Wikipedia

Given a statistic $T$ that is not sufficient, an ancillary complement is a statistic $U$ that is ancillary to $T$ and such that $(T, U)$ is sufficient. Intuitively, an ancillary complement "adds the missing information" (without duplicating any).

I understand that an ancillary statistic wrt a family of possible distributions of the sample, is defined as a statistic whose distribution doesn't depend on any sample distribution in the family of sample distributions.

I was wondering what "a statistic $U$ is ancillary to another statistic $T$" mean, in the definiiton of an ancillary complement?

Thanks and regards!

Answer 1

I'm fairly sure what they meant to say was that $U$ is ancillary for $\theta$, that $(T,U)$ is sufficient for $\theta$, & that $U$ is an ancillary complement to $T$ - where $\theta$ is a population parameter. Like you, I can't see what a statistic being ancillary to another statistic could possibly mean.