  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 $\partial\mathcal{F}$ 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.
