Let $\pi:M\to N$ be a smooth fiber bundle with fiber $F$ and let $g$ be a Riemannian metric on $M$. Using $g$ we may split the tangent bundle of $M$ as $$TM\cong TF\oplus \pi^{\ast}TN,$$ where $TF$ is the tangent bundle along the fiber.
Suppose $F=S^2$ and we can define an almost complex structure $J$ on $M$ by $$J=J_{S^2}\oplus J_{\pi^{\ast}TN},$$ where $J_{S^2}$ is the standard complex structure on the fiber $S^2$ and $J_{\pi^{\ast}TN}\in \mathrm{End}(\pi^{\ast}TN)$ satisfies $J_{\pi^{\ast}TN}^2=1$. Then by construction, a fiber of $\pi$ is a $J$-holomorphic sphere in $(M,J)$.
Now suppose that $\tilde{\pi}:M\to N$ is another $S^2$-bundle which is isomorphic to $\pi$. Then $\tilde{\pi}$ gives rise to another splitting $$TM\cong TF\;\tilde{\oplus} \;\tilde{\pi}^{\ast}TN.$$ Further, suppose we define another almost compelx structure $\tilde{J}$ on $M$ $$\tilde{J}=J_{S^2}\;\tilde{\oplus} \;J_{\tilde{\pi}^{\ast}TN}$$ as above. As above, a fiber of $\tilde{\pi}$ is a $\tilde{J}$-holomorphic sphere.
Questions: How are $J$ and $\tilde{J}$ related? More specifically, is a fiber of $\tilde{\pi}$ also a $J$-holomorphic sphere (and vice-versa)?