A voluminous literature documents that peer review is often biased towards prestigious individuals and institutions and that anonymizing submissions reduces this bias. Yet implementing anonymized review can be difficult — some constituencies resist top-down change and enforcing compliance can be prohibitively expensive. An attractive alternative is giving authors the option to anonymize without enforcing it (“voluntary anonymization”), but does this method actually reduce bias? We answer this question using a quasi-experiment with Institute of Physics Publishing, which adopted a policy encouraging anonymization and rolled it out across its portfolio of 57 physical sciences journals. Examining 156,015 submissions and measuring first author prestige with citations, we found that the policy increased positive peer reviews of low-prestige authors by 2.4% and acceptance by 5.6%. For middle- and high-prestige authors, the policy decreased positive reviews (1.8% and 1% [n.s.]) and final acceptance (4.6% and 2.2% [n.s.]). Voluntary anonymization reduces prestige bias.