We work with Lorentzian pre-length spaces with an anti-Lipschitz time function and use the null distance, whith respect to which the space is complete. Assuming the space has \varepsilon-midpoints resp.\ midpoints $m$ between any timelike related points $p,q$ and such that the $m$ are bounded away from $p,q$ in an appropriate way, that we can construct a (nearly) distance realizer from $p$ to $q$.
This is joint work with Felix Rott.