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Shard integrity seems rather redundant in the current build, as there are rarely more than 5 fights you want to engage in during a single run.
Having the shard integrity cross over between runs could be a good solution to limit a players' opportunities to catch illuvials, but also to push players to catch multiple of the same illuvial to switch out one for the other to give one "time to heal up".
Another option could be to change the ranger's inventory to have the team-slots be limited in integrity. You would be able to upgrade the inventory, and in the process creating another avenue to upgrade, customise and personalise.
These could be tiered from 1-5 and unlocked through progressively more difficult missions in-game such as: Catch 10 illuvials, use 10 common ringnuts, fight 50 battles to unlock the blueprint for Tier 1 inventory slots.
Then you unlock the missions for Tier 2 slots and the missions could be more difficult such as: Catch 15 illuvials, use 5 rare ringnuts, fight 500 fights.
As a benefit and other reason to make the slots they could give an X% boost to the Illuvial's experience gain, or stats during battles or both.