  Unlike their Hermitian counterparts, non-Hermitian (NH) systems may display
an exponential sensitivity to boundary conditions and an extensive number of
edge-localized states in systems with open boundaries, a phenomena dubbed the
"non-Hermitian skin effect." The NH skin effect is one of the primary
challenges to defining a topological theory of NH Hamiltonians, as the
sensitivity to boundary conditions invalidates the traditional bulk-boundary
correspondence. The NH skin effect has recently been connected to the winding
number, a topological invariant unique to NH systems. In this paper, we extend
the definition of the winding number to disordered NH systems by generalizing
established results on disordered Hermitian topological insulators. Our
real-space winding number is self-averaging, continuous as a function of the
parameters in the problem, and remains quantized even in the presence of strong
disorder. We verify that our real-space formula still predicts the NH skin
effect, allowing for the possibility of predicting and observing the NH skin
effect in strongly disordered NH systems. As an application we apply our
results to predict a NH Anderson skin effect where a skin effect is developed
as disorder is added to a clean system, and to explain recent results in
optical funnels.
