alt Hacker NewsExcept Altered Carbon mostly waves the real difficulty away by talking about something like "downloading" a mind into a body.
I don't remember there being anything about growing replacement clones, but it would make sense given the other tech in the story.
The initial murder case Kovacs investigates in both the book/show involves a wealthy man who is killed but has his backup mind inserted into his backup clone so he is “recovered” so the victim has no memory of his killing since the backup was prior to killing instead of optimally just prior to death. There is also a big subplot that is show only with Kovacs’ sister and clones.