i don't think the clay was the reason for the lean…the soil was in adequate for building on and was not amended or removed prior to building…everything i have ever read on the tower states its soil base consisted of clay, silt, and sea shells…none the less….he is not trying to build a tower nor is his structure likely to see any weight…it should only be built in place to catch the floor, not support it…applying an upward force on the floor would be a bad idea, however preventing the floor from deflecting won't harm the floor…
another thing that can be added to the joists is more plywood to fill the web spacing as the builder has already done to a portion of the joists...