Weird teeth-cleaning trick that I discovered from suffering from interstitial cystitis / painful bladder syndrome (IC / BPS): toss a bit of baking soda in your water, such as 1 teaspoon per liter. It makes it alkaline. My teeth got whiter and felt better (less of a film feeling). Baking soda is used as a cooking ingredient and shouldn't be too risky, but don't go overboard... bit of discussion on the IC / BPS aspect…
This is the most effective addition to a mouth cleansing routine I've found. I may apply this wet baking soda "paste" multiple times a day, especially after eating or when I feel a rise in mouth acidity. Also, when I happen to get acid reflux or know I'm going to throw up I pre-emptively keep baking soda in my mouth to protect my teeth from acidity.