Debbie and Alan open a web-based bookstore together. They have been friends for so long that they start their business on a handshake after discussing how they will share both work and profits or losses from the business. Have Debbie and Alan formed a real partnership given that they have signed no written partnership agreement?

Yes

Explanation:

Debbie and Alan have formed a real partnership even though they have signed no written partnership agreement because partnership does not require legal Documentation.

Many partnerships are formed naturally because the people who are involved in the business share similar goals, so their partnerships don't need formation documents to exist.