Family-based heritability estimates of complex traits are often considerably larger than their single-nucleotide polymorphism (SNP) heritability estimates. This discrepancy may be due to non-additive effects of genetic variation, including variation that interacts with other genes or environmental factors to influence the trait. Variance-based procedures provide a computationally efficient strategy to screen for SNPs with potential interaction effects without requiring the specification of the interacting variable. While valuable, such variance-based tests consider only a single trait and ignore likely pleiotropy among related traits that, if present, could improve power to detect such interaction effects. To fill this gap, we propose SCAMPI (Scalable Cauchy Aggregate test using Multiple Phenotypes to test Interactions), which screens for variants with interaction effects across multiple traits. SCAMPI is motivated by the observation that SNPs with pleiotropic interaction effects induce genotypic differences in the patterns of correlation among traits. By studying such patterns across genotype categories among multiple traits, we show that SCAMPI has improved performance over traditional univariate variance-based methods. Like those traditional variance-based tests, SCAMPI permits the screening of interaction effects without requiring the specification of the interaction variable and is further computationally scalable to biobank data. We employed SCAMPI to screen for interacting SNPs associated with four lipid-related traits in the UK Biobank and identified multiple gene regions missed by existing univariate variance-based tests. SCAMPI is implemented in software for public use.