We construct explicit examples of globally regular static, spherically symmetric solutions of general relativity with a phantom scalar field as the source of gravity, describing traversable wormholes with flat asymptotic regions on both sides of the throat as well as regular black holes, in particular, those called black universes. To explain why such phantom fields are not observed under usual conditions, we invoke the concept of “invisible ghosts,” which means that the phantom field decays quickly enough at infinity and is there too weak to be observed. This approach leads to wormhole models in which the spherical radius is almost constant in some range of the radial coordinate near the throat, forming a “long throat.” We discuss the peculiar features and difficulties of the stability problem for such configurations. It is shown that the limiting case of a “maximally long throat” has the form of an unstable model with the Nariai metric. This allows us to conjecture that a long throat does not stabilize wormholes with a scalar source. © 2017, Pleiades Publishing, Ltd.