The study of primes in short intervals goes back to Gauss. Assuming the prime $k$-tuple conjecture, Montgomery and Soundararajan showed that the number of primes in a short moving interval is asymptotically normal. This raises a natural question: What is the joint distribution of weighted prime counts in two or more short intervals? Assuming the Riemann hypothesis and the linear independence conjecture, we show that these weighted counts follow a multivariate Gaussian distribution with weak negative correlations. I will also discuss several consequences, viewed as short-interval analogues of results in the Shanks–Rényi prime number race.