We introduce the concept of a bridge trisection of a neatly embedded surface in a compact four-manifold, generalizing previous work with Alexander Zupan in the setting of closed surfaces in closed four-manifolds. Our main result states that any neatly embedded surface $mathcal{F}$ in a compact four-manifold $X$ can be isotoped to lie in bridge trisected position with respect to any trisection $mathbb{T}$ of $X$. A bridge trisection of $mathcal{F}$ induces a braiding of the link $partialmathcal{F}$ with respect to the open-book decomposition of $partial X$ induced by $mathbb{T}$, and we show that the bridge trisection of $mathcal{F}$ can be assumed to induce any such braiding. We work in the general setting in which $partial X$ may be disconnected, and we describe how to encode bridge trisected surface diagrammatically using shadow diagrams. We use shadow diagrams to show how bridge trisected surfaces can be glued along portions of their boundary, and we explain how the data of the braiding of the boundary link can be recovered from a shadow diagram. Throughout, numerous examples and illustrations are given. We give a set of moves that we conjecture suffice to relate any two shadow diagrams corresponding to a given surface. We devote extra attention to the setting of surfaces in $B^4$, where we give an independent proof of the existence of bridge trisections and develop a second diagrammatic approach using tri-plane diagrams. We characterize bridge trisections of ribbon surfaces in terms of their complexity parameters. The process of passing between bridge trisections and band presentations for surfaces in $B^4$ is addressed in detail and presented with many examples.
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