  While phase transitions between magnetic analogs of three states of matter
--- a long-range ordered state, paramagnet, and spin liquid --- have been
extensively studied, the possibility of "liquid-liquid" transitions, namely,
between different spin liquids, remains elusive. By introducing the additional
Ising coupling into the honeycomb Kitaev model with bond asymmetry, we discover
that the Kitaev spin liquid turns into a spin-nematic quantum paramagnet before
a magnetic order is established by the Ising coupling. The quantum phase
transition between the two liquid states accompanies a topological change
driven by fractionalized excitations, the $Z_2$ gauge fluxes, and is of first
order. At finite temperatures, this yields a persisting first-order transition
line that terminates at a critical point located deep inside the regime where
quantum spins are fractionalized. It is suggested that similar transitions may
occur in other perturbed Kitaev magnets with bond asymmetry.
