  Every synchronising permutation group is primitive and of one of three types:
affine, almost simple, or diagonal. We exhibit the first known example of a
synchronising diagonal type group. More precisely, we show that
$\mathrm{PSL}(2,q)\times \mathrm{PSL}(2,q)$ acting in its diagonal action on
$\mathrm{PSL}(2,q)$ is separating, and hence synchronising, for $q=13$ and
$q=17$. Furthermore, we show that such groups are non-spreading for all prime
powers $q$.
