The win ratio was first proposed in 2012 by Pocock and his colleagues to analyze a composite endpoint while considering the clinical importance order and the relative timing of its components. It has attracted considerable attention since then, in applications as well as methodology. It is not uncommon that some clinical trials require a stratified analysis. In this article, we propose a stratified win ratio statistic in a similar way as the Mantel-Haenszel stratified odds ratio, derive a general form of its variance estimator with a plug-in of existing or potentially new variance/covariance estimators of the number of wins for the two treatment groups, and assess its statistical performance using simulation studies. Our simulations show that our proposed Mantel-Haenszel-type stratified win ratio performs similarly to the Mantel-Haenszel stratified odds ratio for the simplified situation when the win ratio reduces to the odds ratio, and our proposed stratified win ratio is preferred compared to the inverse-variance weighted win ratio and unweighted win ratio particularly when the data are sparse. We also formulate a homogeneity test following Cochran's approach that assesses whether the stratum-specific win ratios are homogeneous across strata, as this method is used frequently in meta-analyses and a better test for the win ratio homogeneity is not available yet.