  We present a suite of cosmological N-body simulations describing the "Running
Friedmann-Lema{\"i}tre-Robertson-Walker" (R-FLRW) cosmological model. This
model is based on quantum field theory in a curved space-time and extends
{\Lambda}CDM with a time-evolving vacuum density, {\Lambda}(z), and
time-evolving gravitational Newton's coupling, G(z). In this paper we review
the model and introduce the necessary analytical treatment needed to adapt a
reference N-body code. Our resulting simulations represent the first
realisation of the full growth history of structure in the R-FLRW cosmology
into the non-linear regime, and our normalisation choice makes them fully
consistent with the latest cosmic microwave background data. The
post-processing data products also allow, for the first time, an analysis of
the properties of the halo and sub-halo populations. We explore the
degeneracies of many statistical observables and discuss the steps needed to
break them. Furthermore, we provide a quantitative description of the
deviations of R-FLRW from {\Lambda}CDM, which could be readily exploited by
future cosmological observations to test and further constrain the model.
