Something I find a little jarring about the game is how the value of your share can vary significantly based on who has the Shaman tile [which specifies a complete tiebreaker based on who has it], and the Shaman tile can change hands without you having further say as to what you get.
So I've tried this and like it:
Don't use the Shaman tile
Instead make tiles numbered 1-5 (enough for your player count)
The numbered tiles don't go in the bag but just sit out in the open.
When you take a share from the common pool, you must take exactly one (unclaimed) numbered tile.
When you take a share from another player, you take their numbered tile and may not throw it back into the pool.
(Therefore, a claimed share always has exactly one numbered tile in it.)
You break ties according to the numbers on the tiles.
So, you can always know where you are in the tiebreaker.
Some would probably find this less interesting/exciting, but the aspect of Mammut that fascinates me is how it seems like you're in control of your fate, and that if you do badly it must be because you picked your shares badly. I just feel the Shaman tiebreaker undermines that feeling.
I like this idea. Kind of combines part of Moon's Oasis into the game.