What are alternatives to toothpaste?

Baking soda,dirt too. Toothpaste without Fluoride has no active therapeutic ingredient. Without micro abrasives staining occurs. Other than taste, you can use baking soda, dirt from your back yard, a slurry of oreo cookies or brush with nothing at all. It doesn't matter what you use anyway as long as you don't harm yourself! in my office: good oral hygiene= floss =93% of home care ; toothbrush 6%; rinse 1%.
Baking soda. Baking soda.
WHY? Why do you want an alternative to toothpaste? Toothpaste helps your toothbrush keep your teeth clean just as soap helps you wash your body. Try to find a toothpaste that you like the taste of. Use small amounts if necessary.
Very few. Why try to re-invent the wheel? Toothpaste is designed for just that... Cleaning and protecting your teeth. Other products can me more abrasive and damaging to your teeth and gums.
Options. Best option is still fluoridated toothpaste, but if you cannot use that, at least look for xylitol pastes and some brush with baking soda. Note, the abrasive coefficient of a toothpaste is also important, especially if you have an acidic mouth.
No need from paste. You can make a paste of baking soda and hydrogen peroxide to brush your teeth. You don't even need toothpaste to brush your teeth. Brushing your teeth disrupts bacteria: toothpaste just helps your mouth feel fresher and "clean".
Water. Toothpaste plays a minor role in the disruption of plaque bacteria from teeth. Careful, detailed brushing of each tooth individually and gently massaging with a wet soft bristle brush will work perfectly (and you can see what you're doing). The Fluoride protection from the paste along with the better taste you apply after the debris and bacteria have been disrupted.