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 randomwalk phase-noise model. A pilot-aided CPE algorithm developed for this model is used to evaluate two strategies, namely jointchannel and per-channel CPE. To quantify the performance differences between the two strategies, their respective phasenoise 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.