In the instrumental variable quantile regression (IVQR) model of Chernozhukov and Hansen (2005), a one-dimensional unobserved rank variable monotonically determines a single potential outcome. Even when multiple outcomes are simultaneously of interest, it is common to apply the IVQR model to each of them separately. This practice implicitly assumes that the rank variable of each regression model affects only the corresponding outcome and does not affect the other outcomes. In reality, however, it is often the case that all rank variables together determine the outcomes, which leads to a systematic correlation between the outcomes. To deal with this, we propose a nonlinear IV model that allows for multivariate unobserved heterogeneity, each of which is considered as a rank variable for an observed outcome. We show that the structural function of our model is locally identified under the assumption that the IV and the treatment variable are sufficiently positively correlated.