The Hubbard model and its strong-coupling version, the Heisenberg one, have been widely studied on the triangular lattice to capture the essential low-temperature properties of different materials. One example is given by transition metal dichalcogenides, as 1T$-$TaS$_2$, where a large unit cell with $13$ Ta atom forms weakly-coupled layers with an isotropic triangular lattice. By using accurate variational Monte Carlo calculations, we report the phase diagram of the $t-t^prime$ Hubbard model on the triangular lattice, highlighting the differences between positive and negative values of $t^prime/t$; this result can be captured only by including the charge fluctuations that are always present for a finite electron-electron repulsion. Two spin-liquid regions are detected: one for $t^prime/t<0$, which persists down to intermediate values of the electron-electron repulsion, and a narrower one for $t^prime/t>0$. The spin-liquid phase appears to be gapless, though the variational wave function has a nematic character, in contrast to the Heisenberg limit. We do not find any evidence for non-magnetic Mott phases in the proximity of the metal-insulator transition, at variance with the predictions (mainly based upon strong-coupling expansions in $t/U$) that suggest the existence of a weak-Mott phase that intrudes between the metal and the magnetically ordered insulator.
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