I was thinking of something like Tether, but it doesn't appear to have much traction
I agree. You'd have to be pretty careful with Tether considering that you have to trust the actions of the Tether company. It goes against BTC's trustless ethos (as does fiat, but this is two levels of it).
One option you could try is moving into gold. There are several companies that accept Bitcoin as payment for gold, including
JM Bullion for example.
You could also try to convert to fiat anonymously by using a peer-to-peer exchange like LocalBitcoins, but of course this would be risky. If you chose to do this, you could try choosing the Cash in Person payment option and opt for people who don't ask for ID.