Setting: Far future, post-space colonization.
I'm not a fan of making up alien languages with a bunch of apostrophes, so the two languages spoken by people in my story are translated into modern English. One language is essentially a lingua franca that all of the characters understand and that corresponds roughly with English in terms of grammar and syntax anyway. The second language is spoken only by a quasi-alien culture that is, among other differences, extremely collectivist in its outlook. These people are forbidden to own anything beyond what they need for personal use (e.g., they can own a toothbrush, but not a house).
This leads to some linguistic issues
When these characters are speaking the lingua franca, their manner of speaking comes off as awkward the minute they start talking about someone else ("the woman who gave birth to me"*). But I don't want their speech to come off as awkward when they're presumably speaking their own language.
So my question: Are there any sort of real-life languages that lack the genitive/possessive case? If so, how do they describe human relationships? And how could you write that sort of grammatical quirk into English without getting excessively wordy?
EDIT: Thanks so much, everyone! You're ridiculously helpful.
* Possibly a bad example, as at least half of them don't know who their biological parents are.