  Recent measurements of the ionizing photon mean free path (MFP) based on
composite quasar spectra may point to reionization ending at $z<6$. These
measurements are challenging because they rely on assumptions about the
proximity zones of the quasars. For example, some quasars might have been close
to neutral patches where reionization was still ongoing ("neutral islands"),
and it is unclear how they would affect the measurements. We address this
question with mock MFP measurements from radiative transfer simulations. We
find that, even in the presence of neutral islands, our mock MFP measurements
agree to within $30~\%$ with the true spatially averaged MFP in our
simulations, which includes opacity from both the ionized medium and the
islands. The inferred MFP is sensitive at the $<~50\%$ level to assumptions
about quasar environments and lifetimes for realistic models. We demonstrate
that future analyses with improved data may require explicitly modeling the
effects of neutral islands on the composite spectra, and we outline a method
for doing this. Lastly, we quantify the effects of neutral islands on
Lyman-series transmission, which has been modeled with optically thin
simulations in previous MFP analyses. Neutral islands can suppress transmission
at $\lambda_{\rm rest}<912$ Angstroms significantly, up to a factor of 2 for
$z_{\rm qso}=6$ in a plausible reionization scenario, owing to absorption by
many closely spaced lines as quasar light redshifts into resonance. However,
the suppression is almost entirely degenerate with the spectrum normalization
and thus does not significantly bias the inferred MFP.
