  Phylogenetic mixtures model the inhomogeneous molecular evolution commonly
observed in data. The performance of phylogenetic reconstruction methods where
the underlying data is generated by a mixture model has stimulated considerable
recent debate. Much of the controversy stems from simulations of mixture model
data on a given tree topology for which reconstruction algorithms output a tree
of a different topology; these findings were held up to show the shortcomings
of particular tree reconstruction methods. In so doing, the underlying
assumption was that mixture model data on one topology can be distinguished
from data evolved on an unmixed tree of another topology given enough data and
the ``correct'' method. Here we show that this assumption can be false. For
biologists our results imply that, for example, the combined data from two
genes whose phylogenetic trees differ only in terms of branch lengths can
perfectly fit a tree of a different topology.
