  Delsarte-Goethals-Seidel showed that if $X$ is a spherical $t$-design with
degree $s$ satisfying $t\geq 2s-2$, $X$ carries the structure of an association
scheme. Also Bannai-Bannai showed that the same conclusion holds if $X$ is an
antipodal spherical $t$-design with degree $s$ satisfying $t=2s-3$. As a
generalization of these results, we prove that a union of spherical designs
with a certain property carries the structure of a coherent configuration. We
derive triple regularity of tight spherical $4,5,7$-designs, mutually unbiased
bases, linked symmetric designs with certain parameters.
