 Solution

An $r_x$ string is a product of X operators, while a $g_z$ string is a product of Z operators. A process that carries $r_x$ once around $g_z$ can be represented by a closed red X string crossing the green Z string once. At the crossing qubit,while all other factors commute. Reversing the order of the two string operators therefore multiplies the state by $-1$. The full braid phase isso $r_x$ and $g_z$ are mutual semions.