The braid groups have two well known Garside presentations. The elegant minimal standard presentation is closely related to the Salvetti complex, a cell complex derived from the complement of the complexification of the real braid arrangement. The dual presentation, introduced by Birman, Ko and Lee, leads to a second Garside structure and a second classifying space, but it has been less clear how the dual braid complex is related to the (quotient of the) complexified hyperplane complement, other than abstractly knowing that they are homotopy equivalent. In this talk, I will discuss recent progress on this issue. Following a suggestion by Daan Krammer, Michael Dougherty and I have been able to embed the dual braid complex into the complement of the complex braid arrangement. This leads in turn to a whole host of interesting complexes, combinatorics, and connections to other parts of the field. This is joint work with Michael Dougherty.