  We show that highly twisted minimal strips can undergo a non-singular
transition, unlike the singular transitions seen in the M\"obius strip and the
catenoid. If the strip is non-orientable this transition is topologically
frustrated, and the resulting surface contains a helical defect. Through a
controlled analytic approximation the system can be mapped onto a scalar
$\phi^4$ theory on a non-orientable line bundle over the circle, where the
defect becomes a topologically protected kink soliton or domain wall, thus
establishing their existence in minimal surfaces. Experimental studies of soap
films confirm these results and demonstrate how the position of the defect can
be controlled through boundary deformation.
