I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.
True, but in the context of a dog, if someone said "Please tie your dog up", I wouldn't think they wanted it wrapped up in chains, I would assume they meant "Please tie your dog's lead to an immovable object", effectively the same as "tethering". However, I agree that "to tether" is not natural for dogs on leads. As Barb said, it sounds as if you're worried your dog will float away unless you secure it to something. Boats are tethered to the dock. Hot air balloons are tethered to the ground.
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BNC quotes sentences in which dogs and cats are being tethered. Please have a look here:
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I understand from your comments that this meaning of "tethered" is obsolete.
Also whenever I tie my dog up (or chain it up) it is because I am afraid that it will float away or run somewhere I don't want it to. Therefore if "tethering" sounds to you as an action taken to prevent the animal from floating away, this is exactly my intended meaning.
You have a floating dog?! Wow! I look forward to reading all about it in Fortean Times.
Remember - correct capitalisation, punctuation and spacing make posts much easier to read.