What is the ruling on tying one's hands behind one's back while walking, and is there any Shar'i evidence prohibiting it?

There is nothing in the Sunnah that prohibits this, although some common people may consider it a fault. If a person abstains from it in environments where it is frowned upon, there is no harm in that. As for a legal prohibition, we know of nothing that indicates it, and the default is that this is among the ordinary matters that are permissible to do.