  An interacting system subjected to a strong linear potential can host a
many-body localized (MBL) phase when being slightly perturbed. This so-called
Wannier-Stark or `tilted-field' MBL phase inherits many properties from the
well-investigated disordered MBL phase, and provides an alternative route to
experimentally engineer interacting localized systems without quenched
disorder. In this work, we investigate the dynamics of entanglement in a
Wannier-Stark MBL system coupled to a dephasing environment. As an accessible
entanglement proxy, we use the third R\'{e}nyi negativity $R_3$, which reduces
to the third R\'{e}nyi entropy in case the system is isolated from the
environment. This measure captures the characteristic logarithmic growth of
interacting localized phases in the intermediate-time regime, where the effects
of the coupling to the environment are not yet dominating the dynamics. Thus it
forms a tool to distinguish Wannier-Stark MBL from non-interacting
Wannier-Stark localization up to intermediate time-scales, and to quantify
quantum correlations in mixed-state dynamics.
