https://www.youtube.com/watch?v=goUfl4x8URc
Essentially all of the permaculture advice for breaking up clay appears to be failing on her hardpan, so she's started using broadforking, which is a sort of compromise between tilling and building soil from the top down. The speaker in this video is being a little more aggressive with the broadfork than most people suggest. The idea is to open the subsoil without blending the upper layers together, resulting in the organic matter-consuming bacterial bloom you get with tilling.
One variant of keylining accomplishes a similar trick on an even smaller scale: you put one long slice into the ground. Water and organic matter seep into that cut and fan out from there.