  We construct the first class of topological solitons in gravity that are
supported by internal electromagnetic flux with vanishing net charges. The
solutions are obtained in a six-dimensional Einstein-Maxwell theory with a
three-form flux, and admit an uplift to type IIB supergravity on T$^4$. They
are asymptotic to a torus fibration over four-dimensional Minkowski spacetime.
An interesting class corresponds to solitons with a BPS particle and its
anti-BPS partner held apart by a vacuum bubble. In type IIB, they correspond to
bound states of BPS and anti-BPS D1-D5 extremal black holes. These metrics are
a particular limit of a larger class of axially symmetric metrics that we
construct and that describe smooth horizonless topological solitons. They
correspond to bound states of three non-BPS bubbles on a line. An important
achievement is that the outer bubbles can carry arbitrary D1-D5 charges that we
can tune to vanishing net charges. We discuss their properties and compare them
to a four-dimensional Schwarzschild black hole of the same mass. We show that
they have a long throat with a large redshift, and that they are ultra-compact
with a characteristic size of 1.52 times the Schwarzschild radius.
