In order of your questions:
1/ Tiger tally (hu fu 虎符) is shaped in that way as a matter of spiritual belief. Around the time tiger tally was first put to use (Xia, Shang, and Zhou dynasties, very much pre-unification), the prevailing belief was that the tiger was the king of the animal and the symbol of might. So using the tiger as a symbol for an item that grants military power is good PR (might of the emperor and all that) and good luck (to put it in modern speak).
2/ Tiger tally is only one type among army tallies ( 兵符 bing fu). The tiger is not the only shape. In later dynasties, army tallies were made in the shape of turtles, fishes, ox, and even bunnies. Some army tallies were made in the shape of a plaque too.
dragon-headed plaque tally
However, because tiger tally is the first shape to be popularly used (and spread during Qin dynasty too), so the name tiger tally became synonymous with army tallies in general.
Fun trivia 1: In Chinese MDZS fanfic, if the story involves Wei Wuxian gaining true control over the Yin Tiger Tally, it’s very popular for him to reforge the tally into 2 smaller bunny-shaped tallies, split between him and Lan Wangji.
(Just an illustration. This looks so cute, doesn’t it?)
Fun Trivia 2: The first few tiger tallies were recorded in history as owned and used by a prince Wei (yes, that Wei) and then a king Wei (still that Wei) in at least two different eras before Qin dynasty. So yes, Wei Wuxian owning an artifact called the Yin Tiger Tally is a historical nod towards the original tiger tally. This is one more mark towards that ‘Wei Ying might actually be long lost royalty’ thing.
3/ Wei Wuxian’ Yin Tiger Tally is shaped like a different kind of tally though, that of a ward. Talisman and tally are written the same way and is essentially the same word in Mandarin Chinese 符 fu (paper wards, amulets, cultivator artifacts, etc…). The Yin Tiger Tally is both an army tally (commanding undead troops) and a cultivator artifact. Wei Wuxian also needs no king’s permission. So that’s why it’s shaped that way and not like the tiger shape you have seen, which is reminiscent of evil warding artifacts used in real life Daoism.
4/ The fact that the tiger halves can become one is all the proof required. In ancient times, tiger tallies require specific forging techniques that were kept secret. The tiger shape is split into 2 halves, a right and a left, with a hidden latch in between. The right half is kept by the emperor. The left half is kept by the army general stationed usually at border provinces, fighting specific enemies.
In ancient times, each general would have their set duties and areas where they must protect. Communication from the crown and the generals occurred without the tiger tally. But usually such communication only involved reports, intelligence, accounting, etc…
Tiger tally is only used for the very serious order: which is an order for the general to move from his designated position and begin a war campaign. This kind of order would typically require a general to move from his designated protected territory too.
When the emperor requires his general to move the army and start a campaign, he will send an envoy bearing written orders (stamped with the emperor seal too). The tiger tally is extra security on top of the royal decree. Only when the 2 halves come together, will the general obey the specific orders in the royal decree.
Tiger tallies are usually made from the same block of materials such as bamboo (very early, in Xia dynasty), iron, jade, or precious stones. With tallies made from bamboo, wood, jade, or stones, the marks on the materials are a type of fingerprint itself. Furthermore, the latch in between the 2 halves will only work if the 2 halves are the genuine article. So the latch is the secret component. On top of that, quote from ancient classics are carved onto the tiger halves. Each tally carries a different classic quote. So the quotes also need to match.
As to how the tiger tallies are used after it’s been verified. For soldiers and commanders beneath the general, his authority is without question. But tiger tallies are only used when the emperor required these generals to move from their territory. That’s where the tallies are used again. Because usually, these generals had designated territory that they could not move from on pains of death (and the death of all of their family members). Keep in mind, these generals held massive military power, so the crown had to keep some measure of control over them. So when a general moved to a different territory, if he could not produce a whole tiger tally, other provincial magistrates and generals would attack them for the crime of treason.
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