  We study the thermodynamical properties of compressed baryonic matter with
strangeness within non-relativistic energy density functional models with a
particular emphasis on possible phase transitions found earlier for a simple
$n,p,e,\Lambda$-mixture. The aim of the paper is twofold: I) examining the
phase structure of the complete system, including the full baryonic octet and
II) testing the sensitivity of the results to the model parameters. We find
that, associated to the onset of the different hyperonic families, up to three
separate strangeness-driven phase transitions may occur. Consequently, a large
fraction of the baryonic density domain is covered by phase coexistence with
potential relevance for (proto)-neutron star evolution. It is shown that the
presence of a phase transition is compatible both with the observational
constraint on the maximal neutron star mass, and with the present experimental
information on hypernuclei. In particular we show that two solar mass neutron
stars are compatible with important hyperon content. Still, the parameter space
is too large to give a definitive conclusion of the possible occurrence of a
strangeness driven phase transition, and further constraints from
multiple-hyperon nuclei and/or hyperon diffusion data are needed.
