We generalize earlier work of Fuertes and Gonzalez-Diez as well
as earlier work of Bauer, Catanese and Grunewald by classifying which of
the irreducible Coxeter groups are (strongly real) Beauville groups. We also
make partial progress on the much more difficult question of which Coxeter
groups are Beauville groups in general as well as discussing the related question
of which Coxeter groups can be used in the construction of mixed Beauville
groups