There is an indirect connection which goes via the representation theory of the symmetric group. The symmetries of the Riemann tensor are equivalent to saying that $R$ transforms according to the two-dimensional irreducible representation of $\mathbb S_4$ corresponding to the partition $[2,2]$. On the other hand let us consider the moduli space $M_{0,4}$ parametrizing four distinct ordered points on the Riemann sphere up to Möbius transformations. The cohomology group $H^1(M_{0,4},\mathbf C)$ is $2$-dimensional, and transforms according to the representation $[2,2]$ under its natural action of $\mathbb S_4$. But we may compute this cohomology group as the space of holomorphic $1$-forms on $M_{0,4}$ with logarithmic poles at infinity. Moreover, the cross-ratio $\chi$ may be considered as a holomorphic function on $M_{0,4}$ and its logarithmic derivative $d \log(\chi)$ is such a 1-form with log poles.

There is an indirect connection which goes via the representation theory of the symmetric group. The symmetries of the Riemann tensor are equivalent to saying that $R$ transforms according to the two-dimensional irreducible representation of $\mathbb S_4$ corresponding to the partition $[2,2]$. On the other hand let us consider the moduli space $M_{0,4}$ parametrizing four distinct ordered points on the Riemann sphere up to Möbius transformations. The cohomology group $H^1(M_{0,4},\mathbf C)$ is $2$-dimensional, and transforms according to the representation $[2,2]$ under its natural action of $\mathbb S_4$. But we may compute this cohomology group as the space of holomorphic $1$-forms on $M_{0,4}$ with at most logarithmic poles at infinity. Moreover, the cross-ratio $\chi$ may be considered as a holomorphic function on $M_{0,4}$ and its logarithmic derivative $d \log(\chi)$ is such a 1-form with log poles. (Note that differentiating your analogue of the Bianchi identity gets rid of the $\pi i$, so we really get something exactly like the usual Bianchi identity!)