It's definitely not as simple as a tag.

Seriously - read Pierson- and what the judges have to say about a mortal wound, and property rights. It's definitely not a "race to the carcass."

Here's what the WashingtonAttorney General's officehas to say on the topic:

GAME ‑- ANIMALS ‑- HUNTING ‑- LEGAL RIGHT OF HUNTERS TO TAG ANIMALS SHOT BY OTHER HUNTERS ‑-CRIMES ‑- LARCENY ‑- TAGGING OF AN ANIMAL BAGGED BY ANOTHER.

1. A hunter who lawfully shoots a game animal acquires a vested legal right in such animal provided he continues to manifest an immediate intention to possess it by exercising actual physical possession and by tagging.

2. Where a second hunter shoots and tags a dying animal bagged by another while the first hunter is attempting to reduce it to possession, the second hunter is guilty of larceny if all other elements of the crime are present.
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So here's a basic example of the premise that a vested legal right attaches as soon as a shot is fired, and an immediate intent to recover/tag is present.