  Looking for the primary rainbow in starlight that is reflected by exoplanets
appears to be a promising method to search for liquid water clouds in
exoplanetary atmospheres. Ice water clouds, that consist of water crystals
instead of water droplets, could potentially mask the rainbow feature in the
planetary signal by covering liquid water clouds. Here, we investigate the
strength of the rainbow feature for exoplanets that have liquid and icy water
clouds in their atmosphere, and calculate the rainbow feature for a realistic
cloud coverage of Earth. We calculate flux and polarization signals of
starlight that is reflected by horizontally and vertically inhomogeneous
Earth--like exoplanets, covered by patchy clouds consisting of liquid water
droplets or water ice crystals. The planetary surfaces are black. On a planet
with a significant coverage of liquid water clouds only, the total flux signal
shows a weak rainbow feature. Any coverage of the liquid water clouds by ice
clouds, however, dampens the rainbow feature in the total flux, and thus the
discovery of liquid water in the atmosphere. On the other hand, detecting the
primary rainbow in the polarization signal of exoplanets appears to be a
powerful tool for detecting liquid water in exoplanetary atmospheres, even when
these clouds are partially covered by ice clouds. In particular, liquid water
clouds covering as little as 10%-20% of the planetary surface, with more than
half of these covered by ice clouds, still create a polarized rainbow feature
in the planetary signal. Indeed, calculations of flux and polarization signals
of an exoplanet with a realistic Earth--like cloud coverage, show a strong
polarized rainbow feature.
