Let $(M,g,I_+,I_-)$ be a compact bi-Hermitian manifold, where $g$ is a Riemannian metric and $I_+$, $I_-$ are two complex structures that are both compatible with $g$. We assume that $I_+$ and $I_-$ commute: $I_+ I_- = I_- I_+$.

For each complex structure, we can define a Lee form. Let $\theta_+$ and $\theta_-$ be the Lee forms corresponding to $I_+$ and $I_-$ respectively. Recall that for a Hermitian manifold $(M,g,I)$ with fundamental 2-form $\omega(X,Y) = g(IX,Y)$, the Lee form $\theta$ is defined by:

$$\theta = -\frac{1}{n-1} \delta \omega \circ I$$

where $\delta$ is the codifferential and $n$ is the complex dimension of $M$.

I'm interested in characterizing bi-Hermitian structures for which $\theta_+ = \theta_-$. In particular, are there known geometric or topological obstructions to the existence of bi-Hermitian structures with $\theta_+ = \theta_-$? Are there any non-trivial examples of bi-Hermitian manifolds satisfying $\theta_+ = \theta_-$ that are not already generalized Kähler?