  Quantum spin ice, modeled for magnetic rare-earth pyrochlores, has attracted
great interest for hosting a U(1) quantum spin liquid, which involves spin-ice
monopoles as gapped deconfined spinons, as well as gapless excitations
analogous to photons. However, the global phase diagram under a [111] magnetic
field remains open. Here we uncover by means of unbiased quantum Monte-Carlo
simulations that a supersolid of monopoles, showing both a superfluidity and a
partial ionization, intervenes the kagom\'e spin ice and a fully ionized
monopole insulator, in contrast to classical spin ice where a direct
discontinuous phase transition takes place. We also show that on cooling,
kagom\'e spin ice evolves towards a valence bond solid similar to what appears
in the associated kagom\'e lattice model [S. V. Isakov et al., Phys. Rev. Lett.
97, 147202 (2006)]. Possible relevance to experiments is discussed.
