I try to construct one.

We know that the normal family is an exponential family, and the natural statistics: sample mean and sample variance are complete, and sufficient for \( \mu \) and \( \sigma^2 \) respectively. By Basu's Theorem any ancillary statistic is independent of the complete sufficient statistic. Since \( \mu \) is a location parameter, and \( \sigma^2 \) is a scale parameter, the statistic

\( \frac {X_1 - X_2} {X_3 - X_4} \sim \text{Cauchy} \) and is ancillary to \( \mu \) and \( \sigma^2 \), which is independent to both sample mean and sample variance.

Of course we are excluding the trivial statistic, e.g. a constant.