Let $(Y_1,\dots, Y_k)$ be a random $k$-tuple uniformly distributed in $(\mathbb R/\mathbb Z)^k$.

Let $X_i = \sum_{j=1}^k i^j Y_j \in \mathbb R/\mathbb Z$.

Then the $X_i$ are $k$-wise independent (and uniformly distributed on $\mathbb R/\mathbb Z$), since the linear map sending $(Y_1,\dots, Y_k)$ to $(X_{i_1},\dots, X_{i_k})$ is given by a $k\times k$ integer matrix with nonzero (Vandermonde) determinant, and such maps preserve the uniform measure on the torus $(\mathbb R/\mathbb Z)^k$.

But we can write $ k! Y_k = \sum_{j=0}^{k-1} (-1)^j \binom{k-1}{j} X_{i-j}$ for any $i$ so $k! Y_k$ is a nonconstant function in the tail sigma-algebra.