How do the concepts of anatta, rebirth and karma coexist together?

Although I have read quite a lot of books about Buddhism, almost all of them were meant for the Western layman so I consider myself to be a beginner compared to most of you here. I hope you can forgive me for possible inaccuracies or inconsistencies in my question. I'm hoping to get answers from Buddhists firmly rooted in the tradition, although of course everyone is most welcome to share their points of view. I have problems coming to terms with the seeming paradox between reincarnation and anatta. How can there be rebirth (which could also include the remembrance of past lives) when there is no soul or inherent self? When searching online most answers refer to all phenomena being interdependent, empty and transient, which is fine, but basically comes down to a sort of Lion King's Circle of Life analogy (beautiful as this still is). But this explanation fails to point out why karma also matters in Buddhism in relation to rebirth. If you are born in an animal's body, you are generally considered to have less karma in stock than when you are born in a happy Buddhist household (or in the Pure Land). How do rebirth and karma rhyme with the concept of anatta? I understand that in your present life you could benefit significantly from pursuing a virtuous life and following the Noble Eightfold Path, and it could have a minute positive change for the world after you're gone, but can you also benefit substantially from living virtuous in your next life after the demise of your current body? If so, how? I really appreciate the Buddhist tradition, but the above concepts currently seem contradictory to me. I hope someone can show me the way out of my maze.