OSHA hasn't adopted Z133, nor do they enforce it. There was a push from TCIA earlier this year for OSHA to create their own standard. No one I know wants that.ANSI standard Z133 2017, for Arborist safety is adopted by OSHA and is the standard you may want to study. It uses the term "secured" rather than tied in so as to ensure a cinching anchor. Devices like the Tree Squeeze have been introduced in order to be in compliance with the standard. A Lineman who climbs poles also need to use a secure connection, not just a belt. I recently watched a training video of this.
But as hunters, we are less experienced and at higher risk.....so it's just common sense. Sticks kick out (and get recalled)... straps break, attachment devices fail... or we could also simply slip. We should never depend on our footing, plus none of that stuff is anywhere near as strong, stable and fault tolerant as an anchor. And nobody is even telling us how strong it is. But there is a quoted Minimum Breaking Strength on our ropes and carabiners. So tie in, manage slack, and have a rappel plan.
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Tree care to OSHA follows OSHA general practices (1910)....for now. There are PPE requirements within 1910 applicable to arb of course.