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.