The performance of pilot-aided joint-channel carrier-phase estimation (CPE) in space-division multiplexed multicore fiber (MCF) transmission with correlated phase noise is studied. To that end, a system model describing uncoded MCF transmission where the phase noise comprises a common laser phase noise, in addition to core- and polarization-specific phase drifts, is introduced. It is then shown that the system model can be regarded as a special case of a multidimensional random-walk phase-noise model. A pilot-aided CPE algorithm developed for this model is used to evaluate two strategies, namely joint-channel and per-channel CPE. To quantify the performance differences between the two strategies, their respective phase-noise tolerances are assessed through Monte Carlo simulations of uncoded transmission for different modulation formats, pilot overheads, laser linewidths, numbers of spatial channels, and degrees of phase-noise correlation across the channels. For 20 GBd transmission with 200 kHz combined laser linewidth and 1% pilot overhead, joint-channel CPE yields up to 3.4 dB improvement in power efficiency or 25.5% increased information rate. Moreover, through MCF transmission experiments, the system model is validated and the strategies are compared in terms of bit-error-rate performance versus transmission distance for uncoded transmission of different modulation formats. Up to 21% increase in transmission reach is observed for 1% pilot overhead through the use of joint-channel CPE.