  The visibility of LyA emitting galaxies during the Epoch of Reionization is
controlled by both diffuse HI patches in large-scale bubble morphology and
small-scale absorbers. To investigate the impact on LyA photons, we apply a
novel combination of analytic and numerical calculations to three scenarios:
(i) the `bubble' model, where only diffuse HI outside ionized bubbles is
present; (ii) the `web' model, where HI exists only in overdense self-shielded
gas; and (iii) the more realistic 'web-bubble' model, which contains both. Our
analysis confirms that there is a degeneracy between the ionization structure
of the intergalactic medium (IGM) and the HI fraction inferred from LyA
surveys, as the three models suppress LyA flux equally with very different HI
fractions. We argue that a joint analysis of the LyA luminosity function and
the rest-frame equivalent width distribution/LyA fraction can break this
degeneracy and provide constraints on the reionization history and its
topology. We further show that constraints can improve if we consider the full
shape of the M_UV-dependent redshift evolution of the LyA fraction of Lyman
break galaxies. Contrary to conventional wisdom, we find that (i) a drop of LyA
fraction larger for UV-faint than for UV-bright galaxies can be reproduced with
web and web-bubble models and therefore does not provide exclusive evidence of
patchy reionization, and (ii) the IGM-transmission PDF is unimodal for bubble
models and bimodal in web models. We further highlight the importance of
galaxy-absorber cross-correlation. Comparing our models to observations, the
neutral fraction at z~7 is likely to be of order of tens of per cent when
interpreted with bubble or web-bubble models. Alternatively, we obtain a
conservative lower limit ~1% in the web models, if we allow for a drop in the
photoionization rate by a factor of ~100 from the post-reionized universe.
[abridged]

Cosmology and Nongalactic Astrophysics;Astrophysics of Galaxies