When genetic variants in a gene cluster are associated with a disease outcome, the causal pathway from the variants to the outcome can be difficult to disentangle. For example, the chemokine receptor gene cluster contains genetic variants associated with various cytokines. Associations between variants in this cluster and stroke risk may be driven by any of these cytokines. Multivariable Mendelian randomization is an extension of standard univariable Mendelian randomization to estimate the direct effects of related exposures with shared genetic predictors. However, when genetic variants are clustered, due to being located in a single genetic region, a Goldilocks dilemma arises: including too many highly-correlated variants in the analysis can lead to ill-conditioning, but pruning variants too aggressively can lead to imprecise estimates or even lack of identification. We propose multivariable methods that use principal component analysis to reduce many correlated genetic variants into a smaller number of orthogonal components that are used as instrumental variables. We show in simulations that these methods result in more precise estimates that are less sensitive to numerical instability due to both strong correlations and small changes in the input data. We apply the methods to demonstrate the most likely causal risk factor for stroke at the chemokine gene cluster is monocyte chemoattractant protein-1.