Rohrlich has conjectured that the multiplicative relations in $\mathbb{C}^\times / \overline{\mathbb{Q}}^\times$ between values of $\Gamma$ at rational numbers are generated by the multiplication formula and the reflection formula. In conceptual terms, Lang says that $\Gamma$ is an odd punctured distribution on $\mathbb{Q}/\mathbb{Z}$, and that conjecturally, it is universal, see [*Relations de distributions et exemples classiques*][1], Deligne's article mentioned in [Felipe Voloch's answer][2] and [this MO answer][3].

On the other hand, the first Bernoulli polynomial $B_1(x)$ also provides an odd punctured distribution on $\mathbb{Q}/\mathbb{Z}$, and we know it is universal (this follows from the non-vanishing of the Dirichlet $L$-values $L(\chi,1)$ with $\chi$ odd). This leads to a conjectural criterion for an arbitrary product of $\Gamma$-values
\begin{equation*}
\Gamma(x_1)^{n_1} \cdots \Gamma(x_r)^{n_r}
\end{equation*}
with $x_i \in \mathbb{Q} \backslash \mathbb{Z}$ and $n_i \in \mathbb{Z}$, to be an algebraic number times a power of $\sqrt{\pi}$. Namely, just check whether the divisor $X = \sum_{i=1}^r n_i [x_i]$ on $\mathbb{Q}/\mathbb{Z}$ is in the kernel of the Bernoulli distribution.

If it is, then using linear algebra, you will be able to write $X$ as a linear combination of the multiplication and reflection relations, because the Bernoulli distribution is universal. Therefore you will be able to compute the $\Gamma$ product as an explicit algebraic number times a power of $\sqrt{\pi}$. This algebraic number will be a cyclotomic number times a product of fractional powers of prime numbers.

If one excepts the possible simplifications of this algebraic number, all this can be made into an algorithm.

In the same article, Lang also asks whether the multiplication and reflection formulas generate the ideal of *polynomial* relations between $\Gamma$-values over $\overline{\mathbb{Q}}(\sqrt{\pi})$. One could try, similarly as above, to give an explicit conjectural criterion for a given polynomial in $\Gamma$-values to be algebraic.

  [1]: https://www.numdam.org/item/SDPP_1977-1978__19_2_A14_0/
  [2]: https://mathoverflow.net/a/7716/6506
  [3]: https://mathoverflow.net/a/318995/6506