The automorphism group of a graph product with no SIL

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We study the automorphisms of graph products of cyclic groups, a class of groups that includes all right-angled Coxeter and right-angled Artin groups. We show that the group of automorphisms generated by partial conjugations is itself a graph product of cyclic groups providing its defining graph does not contain any separating intersection of links (SIL). In the case that all the cyclic groups are finite, this implies that the automorphism group is virtually $\operatorname{CAT}(0)$; it has a finite index subgroup which acts geometrically on a right-angled building.