  The reverse space-time (RST) Sine-Gordon, Sinh-Gordon and nonlinear
Schr\"odinger equations were recently introduced and shown to be integrable
infinite-dimensional dynamical systems. The inverse scattering transform (IST)
for rapidly decaying data was also constructed. In this paper, IST for these
equations with nonzero boundary conditions (NZBCs) at infinity is presented.
The NZBC problem is more complicated due to the associated branching structure
of the associated linear eigenfunctions. With constant amplitude at infinity,
four cases are analyzed; they correspond to two different signs of nonlinearity
and two different values of the phase at infinity. Special soliton solutions
are discussed and explicit 1-soliton and 2-soliton solutions are found. In
terms of IST, the difference between the RST Sine-Gordon/Sinh-Gordon equations
and the RST NLS equation is the time dependence of the scattering data.
Spatially dependent boundary conditions are also briefly considered.
