  A Nielsen-Olesen vortex usually sits in an environment that expels the flux
that is confined to the vortex, so flux is not present both inside and outside.
We construct vortices for which this is not true, where the flux carried by the
vortex also permeates the `bulk' far from the vortex. The idea is to mix the
vortex's internal gauge flux with an external flux using off-diagonal kinetic
mixing. Such `dark' vortices could play a phenomenological role in models with
both cosmic strings and a dark gauge sector. When coupled to gravity they also
provide explicit ultra-violet completions for codimension-two brane-localized
flux, which arises in extra-dimensional models when the same flux that
stabilizes extra-dimensional size is also localized on space-filling branes
situated around the extra dimensions. We derive simple formulae for observables
such as defect angle, tension, localized flux and on-vortex curvature when
coupled to gravity, and show how all of these are insensitive to much of the
microscopic details of the solutions, and are instead largely dictated by
low-energy quantities. We derive the required effective description in terms of
a world-sheet brane action, and derive the matching conditions for its
couplings. We consider the case where the dimensions transverse to the bulk
compactify, and determine how the on- and off-vortex curvatures and other bulk
features depend on the vortex properties. We find that the brane-localized flux
does not gravitate, but just renormalizes the tension in a magnetic-field
independent way. The existence of an explicit UV completion puts the effective
description of these models on a more precise footing, verifying that
brane-localized flux can be consistent with sensible UV physics and resolving
some apparent paradoxes that can arise with a naive (but commonly used)
delta-function treatment of the brane's localization within the bulk.
