  Recent experiments on the Co-based $d^7$ honeycomb materials
BaCo$_2$(AsO$_4$)$_2$ and BaCo$_2$(PO$_4$)$_2$ have drawn renewed interest to
XXZ models with competing interactions beyond the nearest neighbor exchange.
While the case of dominant antiferromagnetic exchange has been extensively
studied, the actual materials exhibit a ferromagnetic nearest neighbor
exchange. Here we show that such a sign change of the dominant nearest neighbor
coupling has substantial consequences on the phase diagram. In particular, the
nature of the quantum disordered phase of interest changes from a plaquette
valence bond crystal to a long-range entangled spin liquid phase. By means of
complementary numerical simulations, based on exact diagonalization and the
pseudo-fermion functional renormalization group, we demonstrate the formation
of a gapless spin liquid state at the heart of the ferromagnetic phase diagram
in the isotropic Heisenberg limit, which gives way to out-of-plane Ising
magnetic order upon inclusion of XXZ anisotropy. The magnetically ordered
phases surrounding this quantum disordered region are much less sensitive to
the sign change of the dominant interaction and can be understood from their
classical analogs. We briefly comment on the relevance of our results for
Co-based $d^7$ honeycomb materials.
