Lead rope
The lead rope is the rope or strap that clips to the halter to lead the horse.

What is it for?
It gives the handler grip to lead the horse or tie it short.
More about this item
Lead ropes come in cotton, nylon or horsemanship rope, with a plain snap or a panic clip. Length is usually around two to four metres.
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Sources: British Horse Society · The Pony Club (UK). Descriptive only; not care or training advice.