)$ is normal precisely when the following is satisfied: a surjective homomorphism $\pi:{\operatorname{\mathcal{NT}}}(X)\to B$ is injective if and only if $\pi$ is injective on ${\operatorname{\mathcal{NT}}}(X)^\delta$ and there is a coaction $\beta$ on $B$ such that $\pi$ is $\delta-\beta$ equivariant.  Apply Corollary \[cor:characterise-normal\].  Let $(G,P)$ be a quasi-lattice ordered group and $X$ a $\tilde\phi$-injective compactly aligned product system over $P$ of $C^*$-correspondences over $A$. The coaction $({\operatorname{\mathcal{NO}}_{\!X}}