  We study the dynamics of a one-dimensional Rydberg lattice gas under
facilitation (anti-blockade) conditions which implements a so-called
kinetically constrained spin system. Here an atom can only be excited to a
Rydberg state when one of its neighbors is already excited. Once two or more
atoms are simultaneously excited mechanical forces emerge, which couple the
internal electronic dynamics of this many-body system to external vibrational
degrees of freedom in the lattice. This electron-phonon coupling results in a
so-called phonon dressing of many-body states which in turn impacts on the
facilitation dynamics. In our theoretical study we focus on a scenario in which
all energy scales are sufficiently separated such that a perturbative treatment
of the coupling between electronic and vibrational states is possible. This
allows to analytically derive an effective Hamiltonian for the evolution of
consecutive clusters of Rydberg excitations in the presence of phonon dressing.
We analyze the spectrum of this Hamiltonian and show -- by employing Fano
resonance theory -- that the interaction between Rydberg excitations and
lattice vibrations leads to the emergence of slowly decaying bound states that
inhibit fast relaxation of certain initial states.
