With difficulty and high costs and secretively. Successful ones are the ones that are able to do it more cheaply. This is also the reason for their secretiveness: prices would go up.
The costly but straightforward approach would be to buy equity index puts. However, I don't think anyone here can or will explain how you can tail hedge at scale significantly more cheaply and effectively. To support this negative answer, I offer this:
In May 2020, there was a Twitter fight between Nassim Taleb (of Black Swan and tail hedging fame) and Cliff Asness (co-founder of 143 billion hedge fund AQR) on this very subject. The fight started after a report by AQR on their blog claimed the strategy happened to work in March 2020 during the (first?) COVID-19 sell-of but was expensive to maintain during all the periods before this. They conclude that over longer periods, buying portfolio insurance isn't a good investment.
Taleb claims that tail-risk hedging can be done and should be done but I'm not aware of him or anyone else giving a general guide on how they exactly do it or point out where the AQR research paper went wrong.
What you can, is do an analysis (company, scenario, macro or something else) and try to figure out where tail risks are underpriced in the market and buy the protection (for example John Paulson's bet against the housing market). In my opinion, this has more in common with active investing than hedging.