I understand the notion of a nontrivial fiber bundle with fiber $F$ over a base manifold $B$, as defined in terms of the projection map $\pi$: for any sufficiently small region $U \subset B$, the preimage $\pi^{-1}(U)$ is homeomorphic to the product space $U \times F$, but the preimage $\pi^{-1}(B)$ itself (the total space) is not homeomorphic to $B \times F$. The Mobius strip is a standard example for visual intuition.

However, physicists like myself often think of a fiber bundle in terms of its sections rather than its projection map. Is there an equivalent definition of a nontrivial bundle formulated in terms of its sections $\sigma$ (the right-inverses of $\pi$)? I.e. a statement of the form "a fiber bundle is nontrivial iff (some section $\sigma$ has)/(all sections $\sigma$ have) property $X$"? If not, is there any intuition for what the sections of a nontrivial bundle "look like"? I know that a principle bundle is nontrivial iff it does not admit any global section, but I'm curious how things work for general fiber bundles.