Current cancer chemotherapies are limited by the lack of tumor-specific targets, which would allow for selective eradication of malignant cells without affecting healthy tissues. In contrast to normal cells, most tumor cells contain multiple centrosomes, which tend to cause the formation of multipolar mitotic spindles, chromosome segregation defects, and cell death. Nevertheless, many cancer cells divide successfully because they can cluster multiple centrosomes into two spindle poles. Inhibition of this centrosomal clustering, with consequent induction of multipolar spindles and subsequent cell death, would specifically target cancer cells and overcome one limitation of current cancer treatments. We have performed a genome-wide RNA interference screen to identify proteins involved in the prevention of spindle multipolarity in human cancer cells with supernumerary centrosomes. The chromosomal passenger complex, Ndc80 microtubule-kinetochore attachment complex, sister chromatid cohesion, and microtubule formation via the augmin complex were identified as necessary for centrosomal clustering. We show that spindle tension is required to cluster multiple centrosomes into a bipolar spindle array in tumor cells with extra centrosomes. These findings may explain the specificity of drugs that interfere with spindle tension for cancer cells and provide entry points for the development of therapeutics.