  Context. A large fraction of the interstellar medium can be characterized as
a multiphase medium. The neutral hydrogen gas is bistable with a cold and warm
neutral medium (CNM and WNM respectively) but there is evidence for an
additional phase at intermediate temperatures, a lukewarm neutral medium (LNM)
that is thermally unstable. Aims. We use all sky data from the HI4PI survey to
separate these neutral HI phases with the aim to determine their distribution
and phase fractions in the local interstellar medium. Methods. HI4PI
observations, gridded on an nside = 1024 HEALPix grid, were decomposed into
Gaussian components. From the frequency distribution of the velocity
dispersions we infer three separate linewidth regimes. Accordingly we extract
the HI line emission corresponding to the CNM, LNM, and WNM. We generated
all-sky maps of these phases in the local HI gas with -8 < v_LSR < 8 km/s.
Results. Each of the HI phases shows distinct structures on all scales. The LNM
never exists as a single phase but contributes on average 41% of the HI. The
CNM is prominent only for 22% of the sky, contributes there on average 34% but
locally up to 60% of the HI and is associated with dust at temperatures T_dust
~ 18.6 K. Embedded cold filaments show a clear anti-correlation between CNM and
LNM. Also the smoothly distributed WNM is anti-correlated with the CNM. It
contributes for the rest of the sky 39% with dust associated at temperatures
T_dust ~ 19.4 K. Conclusions. The CNM in filaments exists on small scales. Here
the observed anti-correlation between LNM and CNM implies that both, filaments
and the surrounding more extended LNM, must have a common origin.
