Recent TeV observations of nearby pulsars with the HAWC telescope have been interpreted as evidence that diffusion of high-energy electrons and positrons within pulsar wind nebulae is highly inefficient compared to the rest of the interstellar medium. If the diffusion coefficient well outside the nebula is close to the value inferred for the region inside the nebula, high-energy electrons and positrons produced by the two observed pulsars could not contribute significantly to the local measured cosmic-ray flux. The HAWC collaboration thus concluded that, under the assumption of isotropic and homogeneous diffusion, the two pulsars are ruled out as sources of the anomalous high-energy positron flux. Here, we argue that since the diffusion coefficient is likely not spatially homogeneous, the assumption leading to such conclusion is flawed. We solve the diffusion equation with a radially dependent diffusion coefficient, and show that the pulsars observed by HAWC produce potentially perfect matches to the observed high-energy positron fluxes. We also study the implications of inefficient diffusion within pulsar wind nebulae on Galactic scales, and show that cosmic rays are likely to have very long residence times in regions of inefficient diffusion. We describe how this prediction can be tested with studies of the diffuse Galactic emission. possibility has received considerable interest ever since the PAMELA result, but is very tightly constrained by the fact that no associated signal has been observed in gamma rays from dark matter annihilation, for example, in local dwarf spheroidal galaxies [16] or in nearby clusters of galaxies [17] For many years, an alternate, more mundane explanation has also been put forward [5,[18][19][20][21][22][23]23]: that middle-aged (τ ∼ 10 5 − 10 6 yr), nearby pulsars could accelerate, in their magnetosphere and, subsequently, in the surrounding shock with the interstellar medium, (primary) electrons and positrons to very high energy. The energetics of known candidate pulsars, as indicated by the pulsars' spin-down luminosity, is in the correct range to explain the observed excess positrons, as long as (1) the diffusion coefficient is D ∼ D ISM between the pulsar and Earth, and (2) a fraction of O(10%) of the spin-down luminosity is injected in positron-electron pairs [20,23].Recently, the HAWC (High-Altitude Water Cherenkov) Observatory confirmed earlier results from Milagro [24] and HAWC [25] and observed extended TeV emission surrounding two nearby pulsars, Geminga and Monogem (PSR B0656+14), among the candidate sources for the observed anomalous highenergy positrons [26]. As previously noted in [27], the spectrum and morphology from TeV gamma-ray observations can be used to infer features of the underlying high-energy electron-positron population responsible for the up-scattering of photons to TeV energies. In fact, Ref. [27] argued that HAWC observations available at the time [25] were entirely compatible, and actually supported, the hypothesis that the observed excess high-energy...