An alternating Mizuno formula, modified gamma functions, and double sine special values

For $n\ge1$ and $\operatorname{Re}(x_j)>0$, we prove an alternating analogue of Mizuno's product formula: \begin{equation*} \prod\nolimits_{m=0}^{\infty} \left(\prod_{j=1}^{n}(m+x_j)\right)^{(-1)^m} = \frac{(\sqrt{π/2})^n} {\prod_{j=1}^{n}\widetildeΓ(x_j)}, \end{equation*} where $\widetildeΓ$ is the