I need to generate a set of random points on an unit sphere, using Mathematica. I can do it for a simple homogeneous (i.e. uniform) distribution of points using the following coordinates :

RandomPoints[u_, phi_] := {Sqrt[1 - u^2] Cos[phi], Sqrt[1 - u^2] Sin[phi], u},

with uniform random variables u in [-1, 1] and phi in [0, 2 Pi].

However, I would like to define a distribution that feels more "natural", i.e. Perlin-like (take note that I don't know how to define "Perlin noise", mathematically).

What do you suggest ?