  We present complete simplicial fan realizations of any spherical subword
complex of type $A_n$ for $n\leq 3$. This provides complete simplicial fan
realizations of simplicial multi-associahedra $\Delta_{2k+4,k}$, whose facets
are in correspondence with $k$-triangulations of a convex $(2k+4)$-gon. This
solves the first open case of the problem of finding fan realizations where
polytopality is not known. The techniques presented in this paper work for all
finite Coxeter groups and we hope that they will be useful to construct fans
realizing subword complexes in general. In particular, we present fan
realizations of two previously unknown cases of subword complexes of type
$A_4$, namely the multi-associahedra $\Delta_{9,2}$ and $\Delta_{11,3}$.
