  We present a systematic approach for the semiclassical treatment of many-body
dynamics of interacting, open spin systems. Our approach overcomes some of the
shortcomings of the recently developed discrete truncated Wigner approximation
(DTWA) based on Monte-Carlo sampling in a discrete phase space that improves
the classical treatment by accounting for lowest-order quantum fluctuations. We
provide a rigorous derivation of the DTWA by embedding it in a continuous phase
space, thereby introducing a hybrid discrete-continuous truncated Wigner
approximation (DCTWA). We derive a set of operator-differential mappings that
yield an exact equation of motion (EOM) for the continuous SU(2) Wigner
function of spins. The standard DTWA is then recovered by a systematic
neglection of specific terms in this exact EOM. The hybrid approach permits us
to determine the validity conditions and to gain detailed understanding of the
quality of the approximation, paving the way for systematic improvements.
Furthermore, we show that the continuous embedding allows for a straightforward
extension of the method to open spin systems subject to dephasing, losses and
incoherent drive, while preserving the key advantages of the discrete approach,
such as a positive definite Wigner distribution of typical initial states. We
derive exact stochastic differential equations for processes which cannot be
described by the standard DTWA due to the presence of non-classical noise. We
illustrate our approach by applying it to the dissipative dynamics of Rydberg
excitation of one-dimensional arrays of laser-driven atoms and compare it to
exact results for small systems.
