A necessary condition for an example is that there are positive integers $a,b$ such that $\prod_{i=0}^{2N-1}(a+ib)$ is a square. Differently phrased (divide by $b^{2N}$), we ask for rational solutions of the hyperelliptic curve $Y^2=\prod_{i=0}^{2N-1}(X+i)$. According to a conjecture by Sander (see [Nontrivial rational points on Erdős-Selfridge curves][1]), there are no solutions except a trivial one which does not meet your positivity requirement.


  [1]: https://arxiv.org/abs/2411.05221