  We consider the possibility that tunneling in a degenerate double-well
potential in de Sitter spacetime leads to coherent oscillations of quantum
probability to find the system in a given well. We concentrate on the case when
the mass scale of the potential is much larger than the Hubble parameter and
present a procedure for analytical continuation of the ``time''-dependent
instantons, which allows us to study the subsequent real-time evolution. We
find that the presence of the de Sitter horizon makes tunneling completely
incoherent and calculate the decoherence time. We discuss the difference
between this case and the case of a $\theta$-vacuum, when tunneling in de
Sitter spacetime preserves quantum coherence.
