Gift not adding
Check the cart subtotal vs the tier threshold — is the cart actually above the goal amount in the customer's currency?
Check the gift variant is in stock.
Check the unified discount function is deployed (Settings → Discounts).
Gift not removing when cart drops
Force-refresh the cart.
If it persists, the function isn't deployed properly. Re-deploy from Settings → Discounts.
Multiple gifts when only one was expected
Check your tier stacking toggle. By default, only the highest qualified tier of each reward type applies; if cumulative tiers is enabled, multiple gifts can stack.
Gift shows in cart but not at checkout
Almost always a combinations rule. Open Discounts → Combinations in Shopify admin and verify your active codes allow combining with product / shipping discounts.
Still stuck?
Run Validate Discount with the customer's exact cart to see what the Function thinks. Share the output with support.