Start with the 2-sphere $\mathcal{S}^2$ with the standard $(\theta, \phi)$ chart ($\theta = 0$ is the North pole etc) and metric:
$$ds^2 = d\theta^2 + \sin^2 \theta d\phi^2$$
Remove the two poles from $\mathcal{S}^2$ and the arc $\phi=\pi$ to get $\mathcal{S}^2_0$. Let $A:=(0, \pi) \times (-\pi, \pi) \subset \mathbb{R}^2$. Next define the standard diffeomorphism $f: A \rightarrow \mathcal{S}^2_0$ between the two manifolds and let $f^*(T\mathcal{S}^2_0)$ be the corresponding bundle-pullback of the Tangent Bundle of $\mathcal{S}^2_0$. Construct the connection $\nabla_T$ on $\mathcal{S}^2_0$ (T for Torsion) where $\Gamma_{\theta \phi}^{\phi} = \cot\theta$ and all the other connection coefficients are zero.
It can be shown that $\nabla_T$ is metric compatible and flat (ie. of zero curvature) but it has Torsion. One can also define the connection-pullback of $\nabla_T$ which is essentially the unique connection on $A \subset \mathbb{R}^2$, call it $\nabla$, induced by the Tangent Bundle of $\mathcal{S}^2_0$ via its bundle-pullback. Can we picture this connection $\nabla$ on $A$? More generally:
Can we visualize the non-zero Torsion of the pair ($\mathcal{S}^2_0$, $\nabla_T$) via the "flat" pair $(A, \nabla)$?
I am trying to gain geometric intuition so I can ultimately answer this question on the relation between fibers and Torsion. Concrete calculations are of course welcome and they can add to the insight but they are secondary to me. Huge thanks to @Moishe Kohan and @Ted Shifrin for helping me with the phrasing of this question.
