Generating "amoeba-like" clusters around lat/lng points

I have a fairly unique problem I'm trying to solve, and the math is confusing me. I'm working on a map project, and one of the needs is to generate random, "amoeba-like" clusters of points around a given latitude/longitude center.

By "amoeba-like", I mean the cluster shouldn't be overly round or uniform, but should, ideally, have "arms" that extend out from the center. At the same time, the arms should be somewhat random and not particularly spiral-like.

I've tried a number of approaches, but so far, none of the clusters I've generated are quite the shape I want. I've tried approaches similar to what's described here, and I've tried the make_blobs from sklearn. But I couldn't figure out how to extend either of those get the "arms" I'm looking for, and I'm not confident that either of those approaches really makes sense.

Any advice anyone could provide would be much appreciated!