Multi-connectivity (MC) is regarded as one of the key features that meet the requirements of ultra-reliable low-latency communications for 5th generation networks, as it provides multiple diversity branches. Recently, we evaluated the decoding reliability of various MC setups and different combining algorithms by analyzing the resulting outage probabilities. In this work, we are interested in how much the performance is affected by correlated fading. Specifically, we analyze the outage probability of MC systems with joint decoding reception operating over correlated quasi-static Rayleigh and Nakagami- {m} fading channels. Our main contributions are as follows: (i) deriving the exact outage probability in integral form and the asymptotic outage probability at high signal-to-noise ratio (SNR) in closed form; (ii) deriving the correlation loss, which quantifies the extra SNR required under correlated fading as compared with the independent scenario; and (iii) evaluating frame-error rates of quasi-cyclic low-density parity-check codes by Monte-Carlo simulations. Our results show that the correlation loss is marginal for a low to moderate level of correlation, whereas the diversity gain is not affected by correlation whatsoever. Furthermore, we find that the correlation loss is independent of the degrees of freedom encapsulated by the Nakagami- {m} fading parameter.