We obtain general coincidence theorems and related results for multimaps in very large classes defined on $\omega$-connected spaces. Our typical consequence is as follows: Let $X$ be a compact $\omega$-connected topological space, and $F : X \multimap X$ a multimap with nonempty values and open fibers such that, for each open subset $O \subset X$, $\bigcap_{x \in O} Fx$ is empty or $\omega$-connected. Then $F$ has a fixed point.