Let [katex]\Omega^+\subset\mathbb R^{n+1}[/katex] be an NTA domain and let [katex]\Omega^-= \mathbb R^{n+1}\setminus \overline{\Omega^+}[/katex] be an NTA domain as well. Denote by [katex]\omega^+[/katex] and [katex]\omega^-[/katex] their respective harmonic measures. Assume that [katex]\Omega^+[/katex] is a [katex]\delta[/katex]-Reifenberg flat domain, for some [katex]\delta>0[/katex] small enough. In this paper we show that [katex]\log\frac{d\omega^-}{d\omega^+}\in VMO(\omega^+)[/katex] if and only if [katex]\Omega^+[/katex] is vanishing Reifenberg flat, [katex]\omega^+[/katex] has big pieces of uniformly rectifiable measures, and the inner unit normal of [katex]\Omega^+[/katex] 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.