We initiate the study of outer automorphism groups of special groups G, in the Haglund-Wise sense. We show that Out(G) is infinite if and only if G splits over a subgroup of a centraliser and there exists an infinite-order "generalised Dehn twist". Similarly, the coarsemedian preserving subgroup Outcmp(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 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.