In research synthesis, publication bias (PB) refers to the phenomenon that the publication of a study is associated with the direction and statistical significance of its results. Consequently, it may lead to biased (commonly optimistic) estimates of treatment effects. Visualization tools such as funnel plots have been widely used to investigate PB in univariate meta‐analyses. The trim and fill procedure is a nonparametric method to identify and adjust for PB. It is popular among applied scientists due to its simplicity. However, most visualization tools and PB correction methods focus on univariate outcomes. For a meta‐analysis with multiple outcomes, the conventional univariate trim and fill method can only account for different outcomes separately and thus may lead to inconsistent conclusions. In this article, we propose a bivariate trim and fill procedure to simultaneously account for PB in the presence of two outcomes that are possibly associated. Based on a recently developed galaxy plot for bivariate meta‐analysis, the proposed procedure uses a data‐driven imputation algorithm to detect and adjust PB. The method relies on the symmetry of the galaxy plot and assumes that some studies are suppressed based on a linear combination of outcomes. The method projects bivariate outcomes along a particular direction, uses the univariate trim and fill method to estimate the number of trimmed and filled studies, and yields consistent conclusions about PB. The proposed approach is validated using simulated data and is applied to a meta‐analysis of the efficacy and safety of antidepressant drugs.