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As DaveP said, there are two possible solutions .....

I was puzzling upon why I had to be careful where I stuck the string on the globe to come up with something believable when I realised that I wasn’t being bold enough.  How about this? To explain it without maths. Think of two wire rings lying on the surface of a globe, one a small circle of equal zenith distance (an ‘isozend’) and the other a meridian and its opposite.  If the two rings cross in one place, they must cross again somewhere else, even if it’s on the far side of the Earth and below the observer’s visual horizon.  DaveP