  We investigate recent claims concerning a new class of cosmic string
solutions in the Weinberg-Salam model. They have the general form of previously
discussed semi-local and electroweak strings, but are modified by the presence
of a non-zero W-condensate in the core of the string. We explicitly construct
such solutions for arbitrary values of the winding number $N$. We then prove
that they are gauge equivalent to bare electroweak strings with winding number
$N-1$. We also develop new asymptotic expressions for large-$N$ strings.
