  We show that a cosmic string associated with spontaneous $U(1)_R$ symmetry
breaking gives a constraint for supersymmetric model building. In some models,
the string can be viewed as a tube-like domain wall with a winding number
interpolating a false vacuum and a true vacuum. Such string causes
inhomogeneous decay of the false vacuum to the true vacuum via rapid expansion
of the radius of the tube and hence its formation would be inconsistent with
the present Universe. However, we demonstrate that there exist metastable
solutions which do not expand rapidly. Furthermore, when the true vacua are
degenerate, the structure inside the tube becomes involved. As an example, we
show a "bamboo"-like solution, which suggests a possibility observing an
information of true vacua from outside of the tube through the shape and the
tension of the tube.
