For the directed landscape, the putative universal space-time scaling limit object in the (1+1) dimensional Kardar-Parisi-Zhang (KPZ) universality class, consider the geodesic tree -- the tree formed by the coalescing semi-infinite geodesics in a given direction. As shown in Bhatia '23, this tree comes interlocked with a dual tree, which (up to a reflection) has the same marginal law as the geodesic tree. Analogous examples of one ended planar trees formed by coalescent semi-infinite random paths and their duals are objects of interest in various other probability models, a classical example being the Brownian web, which is constructed as a scaling limit of coalescent random walks. In this paper, we continue the study of the geodesic tree and its dual in the directed landscape and exhibit a new space-filling curve traversing between the two trees that is naturally parametrized by the area it covers and encodes the geometry of the two trees; this parallels the construction of the Tóth-Werner curve between the Brownian web and its dual. We study the regularity and fractal properties of this Peano curve, exploiting simultaneously the symmetries of the directed landscape and probabilistic estimates obtained in planar exponential last passage percolation, which is known to converge to the directed landscape in the scaling limit. On the way, we develop a novel coalescence estimate for geodesics, and this has recently found application in other work.