A random multiplicative function is a (completely) multiplica- tive sequence f (1), f (2), . . . such that the f (p) for primes p are independent random variables uniformly distributed on either the unit circle (known as the Steinhaus model) or the set {−1,+1} (known as the Rademacher model). They are used to model the M ̈obius and/or the related Liouville function, the mean value of which has a very rich structure and contains essential information about the distribution of prime numbers. By work of Harper, it is known that the mean value of a random multiplicative function over a long interval [1,x] does not converge in distribution to a Gaussian with the expected parameters, meaning that multiplicativity interferes just enough with independence. However, Klurman, Shkredov and Xu estab- lished a central limit theorem for the mean value of a Steinhaus random multiplicative function over the image of a polynomial with distinct roots such as n(n+1), say. In ongoing joint work with Jake Chinis, we establish the same result in the more intricate Rademacher model. The techniques we use come from a blend of probability theory as well as analytic and algebraic number theory on counting integral points on ”twists” of certain curves and surfaces.