  We initiate the study of outer automorphism groups of special groups $G$, in
the Haglund-Wise sense. We show that ${\rm Out}(G)$ is infinite if and only if
$G$ splits over a co-abelian subgroup of a centraliser and there exists an
infinite-order "generalised Dehn twist". Similarly, the coarse-median
preserving subgroup ${\rm Out}_{\rm cmp}(G)$ is infinite if and only if $G$
splits over an actual centraliser and there exists an infinite-order
coarse-median-preserving generalised Dehn twist.
  The proof is based on constructing and analysing non-small, stable
$G$-actions on $\mathbb{R}$-trees whose arc-stabilisers are centralisers or
closely related subgroups. Interestingly, tripod-stabilisers can be arbitrary
centralisers, and thus are large subgroups of $G$.
  As a result of independent interest, we determine when generalised Dehn
twists associated to splittings of $G$ preserve the coarse median structure.
