Let \Omega^+\subset\mathbb R^{n+1} be an NTA domain and let \Omega^-= \mathbb R^{n+1}\setminus \overline{\Omega^+} be an NTA domain as well. Denote by \omega^+ and \omega^- their respective harmonic measures. Assume that \Omega^+ is a \delta-Reifenberg flat domain, for some \delta>0 small enough. In this paper we show that \log\frac{d\omega^-}{d\omega^+}\in VMO(\omega^+) if and only if \Omega^+ is vanishing Reifenberg flat, \omega^+ has big pieces of uniformly rectifiable measures, and the inner unit normal of \Omega^+ has vanishing oscillation with respect to the approximate normal. This result can be considered as a two-phase counterpart of a more well known related one-phase problem for harmonic measure solved by Kenig and Toro.