Dressing transformations of constrained Willmore surfaces

We use the dressing method to construct transformations of constrained Willmore surfaces in arbitrary codimension. An adaptation of the Terng--Uhlenbeck theory of dressing by simple factors to this context leads us to define B\"acklund transforms of these surfaces for which we prove Bianchi permutability. Specialising to codimension 2, we generalise the Darboux transforms of Willmore surfaces via Riccati equations, due to Burstall-Ferus-Leschke-Pedit-Pinkall, to the constrained Willmore case and show that they amount to our B\"acklund transforms with real spectral parameter.