  In a recent study, Bryant, Francis and Steel investigated the concept of
\enquote{future-proofing} consensus methods in phylogenetics. That is, they
investigated if such methods can be robust against the introduction of
additional data like extra trees or new species. In the present manuscript, we
analyze consensus methods under a different aspect of introducing new data,
namely concerning the discovery of new clades. In evolutionary biology, often
formerly unresolved clades get resolved by refined reconstruction methods or
new genetic data analyses. In our manuscript we investigate which properties of
consensus methods can guarantee that such new insights do not disagree with
previously found consensus trees but merely refine them. We call consensus
methods with this property \emph{refinement-stable}. Along these lines, we also
study two famous super tree methods, namely Matrix Representation with
Parsimony (MRP) and Matrix Representation with Compatibility (MRC), which have
also been suggested as consensus methods in the literature. While we (just like
Bryant, Francis and Steel in their recent study) unfortunately have to conclude
some negative answers concerning general consensus methods, we also state some
relevant and positive results concerning the majority rule (MR) and strict
consensus methods, which are amongst the most frequently used consensus
methods. Moreover, we show that there exist infinitely many consensus methods
which are refinement-stable and have some other desirable properties.
