  We study the dissipative dynamics of a fermionic superfluid in presence of
two-body losses. We use a variational approach for the Lindblad dynamics and
obtain dynamical equations for Anderson's pseudo-spins where dissipation enters
as a complex pairing interaction as well as effective, density-dependent,
single particle losses which break the conservation of the pseudo-spin norm. We
show that this latter has key consequences on the dynamical behavior of the
system. In the case of a sudden switching of two-body losses we show that the
superfluid order parameter decays much faster than then particle density at
short times and eventually slows-down, setting into a power-law decay at longer
time scales driven by the depletion of the system. We then consider a quench of
pairing interaction, leading to coherent oscillations in the unitary case,
followed by the switching of the dissipation. We show that losses wash away the
dynamical BCS synchronization by introducing not only damping but also a
renormalization of the frequency of coherent oscillations, which depends
non-linearly from the rate of the two-body losses.
