  We determine the degree of fine tuning needed in a generalised version of the
NMSSM that follows from an underlying Z4 or Z8 R symmetry. We find that it is
significantly less than is found in the MSSM or NMSSM and extends the range of
Higgs mass that have acceptable fine tuning up to Higgs masses of mh ~ 130 GeV.
For universal boundary conditions analogous to the CMSSM the phenomenology is
rather MSSM like with the singlet states typically rather heavy. For more
general boundary conditions the singlet states can be light, leading to
interesting signatures at the LHC and direct detection experiments.
