  In 'no common mechanism' (NCM) models of character evolution, each character
can evolve on a phylogenetic tree under a partially or totally separate process
(e.g. with its own branch lengths).
  In such cases, the usual conditions that suffice to establish the statistical
consistency of tree reconstruction by methods such as maximum likelihood (ML)
break down, suggesting that such methods may be prone to statistical
inconsistency (SIN). In this paper we ask whether we can avoid SIN for tree
topology reconstruction when adopting such models, either by using ML or any
other method that could be devised. We prove that it is possible to avoid SIN
for certain NCM models, but not for others, and the results depend delicately
on the tree reconstruction method employed. We also describe the biological
relevance of some recent mathematical results for the more usual 'common
mechanism' setting. Our results are not intended to justify NCM, rather to set
in place a framework within which such questions can be formally addressed.
