  The chiral AIII symmetry class in the periodic table of topological
insulators contains topological phases classified by a winding number $\nu$ for
each odd space-dimension. An open problem for this class is the
characterization of the phases and phase-boundaries in the presence of strong
disorder. In this work, we derive a covariant real-space formula for $\nu$ and,
using an explicit 1-dimensional disordered topological model, we show that
$\nu$ remains quantized and non-fluctuating when disorder is turned on, even
though the bulk energy-spectrum is completely localized. Furthermore, $\nu$
remains robust even after the insulating gap is filled with localized states,
but when the disorder is increased even further, an abrupt change of $\nu$ to a
trivial value is observed. Using exact analytic calculations, we show that this
marks a critical point where the localization length diverges. As such, in the
presence of disorder, the AIII class displays a markedly different physics from
everything known to date, with robust invariants being carried entirely by
localized states and bulk extended states emerging from an absolutely localized
spectrum. Detailed maps and a clear physical description of the phases and
phase boundaries are presented based on numerical and exact analytic
calculations.
