Context. A large fraction of the interstellar medium can be characterized as a multiphase medium. The neutral hydrogen gas is bistable with 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 H I phases with the aim to determine their distribution and phase fractions f 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 H I line emission corresponding to the CNM, LNM, and WNM. We generateed all-sky maps of these phases in the local H I gas with − 8 < vLSR < 8 km s−1. Results. Each of the H I phases shows distinct structures on all scales. The LNM never exists as a single phase but contributes on average 41% of the H I. The CNM is prominent only for 22% of the sky, contributes there on average 34% but locally up to 60% of the H I and is associated with dust at temperatures Tdust ~ 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 Tdust ~ 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.