You can pack as many vectors $m_i$ and $\mu_i$ as you want in $\mathbb{S}^2$. Below I give an example with $5$ points: consider $5$ points $m_i$ forming a regular pentagon parallel to the equator and a bit below it. Below is a figure showing approximately how the north pole $N$ and the five geodesics perpendicular to the $m_i$ look like (note that $\langle N,m_i\rangle<0$ for all $i$).
Now choose $\mu_i$ in the positions of the green points, so that $\langle \mu_i,m_i\rangle>0$ for all $i$ but for $j\neq i$ we have $\langle \mu_i,m_j\rangle<0$.