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In Brownian last-passage percolation (BLPP), the Busemann functions $\mathcalB^{\theta}(\mathbf x,\mathbf y)$ are indexed by two points $\mathbf x,\mathbf y\in \mathbb Z \times \mathbb R$, and a direction parameter $\theta > 0$. Wederive the joint distribution of Busemann functions across all directions. Theset of directions where the Busemann process is discontinuous, denoted$\Theta$, provides detailed information about the uniqueness and coalescence ofsemi-infinite geodesics. The uncountable set of initial points in BLPP givesrise to new phenomena not seen in discrete models. For example, in everydirection $\theta > 0$, there exists a countably infinite set of initial points$\mathbf x$ such that there exist two $\theta$-directed geodesics that splitbut eventually coalesce. Further, we define the competition interface in BLPPand show that the set of initial points whose competition interface isnontrivial has Hausdorff dimension $\frac{1}{2}$. From each of theseexceptional points, there exists a random direction $\theta \in \Theta$ forwhich there exists two $\theta$-directed semi-infinite geodesics that splitimmediately and never meet again. Conversely, when $\theta \in \Theta$, fromevery initial point $\mathbf x \in \mathbb Z \times \mathbb R$, there existstwo $\theta$-directed semi-infinite geodesics that eventually separate.Whenever $\theta \notin \Theta$, all $\theta$-directed semi-infinite geodesicscoalesce.

Global structure of semi-infinite geodesics and competition interfaces in Brownian last-passage percolation