Let $\mathfrak{D}$ be a scattering diagram. Then there is a scattering diagram $\operatorname{{\mathsf{S}}}(\mathfrak{D})$ such that $\operatorname{{\mathsf{S}}}(\mathfrak{D})\setminus\mathfrak{D}$ consists just of rays and $\theta_{\gamma,\operatorname{{\mathsf{S}}}(\mathfrak{D})}$ is the identity for any loop $\gamma$ around the origin.