I am wondering if it is possible for data to be stochastically independent while still having a relationship between the variables?
Just an example let us assume we have data points that have no stochastical independence but we can still find an equation that is relating the variables by $x^2+y^2=r^2$ (points are lying on a circle). I know that points on the circle might be correlated. I just wanted to make clear what I am asking, at least I hope I could make that clear :D.
I am not sure this is what you are looking for, but this example comes to my mind. Suppose that $X_1,\ldots,X_n$ are iid $N(\mu,\sigma^2)$ random variables. Define the sample mean $$ \bar X_n=n^{-1}(X_1+\ldots+X_n) $$ and the sample variance $$ S_n^2=(n-1)^{-1}\sum_{i=1}^n(X_i-\bar X_n)^2. $$ Denote $$ Z=(\bar X_n-\mu)n^{1/2}/\sigma $$ and $$ V=(n-1)S_n^2/\sigma^2. $$ $Z$ and $V$ have $X_1,\ldots,X_n$ in their expressions and it seems that they should be dependent. However, they are independent. The ratio $Z/\sqrt{V/(n-1)}$ has Student's $t$-distribution with $n-1$ degrees of freedom (see here for more details).
I hope this is useful.