Great question — answering as the UphillTech team:
Apparently the behaviour is specific to IOS. I fixed it by not linking directly to the product page on IOS, but to the search page of the product. Like this: https://www.amazon.com/s?k=0099590085 I added the product ID as keyword param. Now your affiliate link will work and when the user click on your affiliate link it will not redirect back to browser (if they have the Amazon app). Just it adds one unnecessary step for your user to have to click on the product, but this is still better than them getting redirected back to the browser. If anyone has a better idea or if it got fixed, please let me know.
Full answer by Fudge: https://stackoverflow.stackexchange.com/a/70106915 (CC BY-SA 4.0).
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