  The well-known problem of wormholes in general relativity (GR) is the
necessity of exotic matter, violating the Weak Energy Condition (WEC), for
their support. This problem looks easier if, instead of island-like
configurations, one considers string-like ones, among them, cylindrically
symmetric space-times with rotation. However, for cylindrical wormhole
solutions a problem is the lacking asymptotic flatness, making it impossible to
observe their entrances as local objects in our Universe. It was suggested to
solve this problem by joining a wormhole solution to flat asymptotic regions at
some surfaces $\Sigma_-$ and $\Sigma_+$ on different sides of the throat. The
configuration then consists of three regions, the internal one containing a
throat and two flat external ones. We discuss different kinds of source matter
suitable for describing the internal regions of such models (scalar fields,
isotropic and anisotropic fluids) and present two examples where the internal
matter itself and the surface matter on both junction surfaces $\Sigma_\pm$
respect the WEC. In one of these models the internal source is a stiff perfect
fluid whose pressure is equal to its energy density, in the other it is a
special kind of anisotropic fluid. Both models are free from closed timelike
curves. We thus obtain examples of regular twice asymptotically flat wormhole
models in GR without exotic matter and without causality violations.
