This app was mentioned in 2 comments, with an average of 2.00 upvotes
We use "denkst du daran". It syncs almost immediately via push messages, has a clean layout so it's easy to read while shopping and can give you notification when someone else added an item. To sync between devices you've to enter the same email address in the app on each device. On Samsung devices it also supports the multi-window feature which can be useful if you want to have a recipe and the shopping list app open at the same time.
Android: https://play.google.com/store/apps/details?id=de.piobyte.android.ddd iOS: https://itunes.apple.com/de/app/denkst-du-daran-einkaufsliste/id616041713
Google Keep works great too, but I don't like the size of the checkboxes and font for this use-case. While using it one handed during shopping the UI shouldn't be too small (imho)
I use this one and love it. If you enter the same email adress for syncing in both installations it will sync almost instantly between them and also notifiy by notifications when new stuff got added to the list
https://play.google.com/store/apps/details?id=de.piobyte.android.ddd