Firms' algorithm development practices are often homogenous. Whether firms train algorithms on similar data, aim at similar benchmarks, or rely on similar pre-trained models, the result is correlated predictions. We model the impact of correlated algorithms on competition in the context of personalized pricing. Our analysis reveals that (1) higher correlation diminishes consumer welfare and (2) as consumers become more price sensitive, firms are increasingly incentivized to compromise on the accuracy of their predictions in exchange for coordination. We demonstrate our theoretical results in a stylized empirical study where two firms use personalized pricing algorithms to determine consumers' willingness to pay. Our results underscore the potential anti-competitive effects of algorithmic pricing and highlight the need for refined antitrust approaches in the era of digital markets.