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In many social science experiments, subjects often interact with each other and as a result one unit's treatment influences the outcome of another unit. Over the last decade, a significant progress has been made toward causal inference in the presence of such interference between units. Researchers have shown that the two-stage randomization of treatment assignment enables the identification of average direct and spillover effects. However, much of the literature has assumed perfect compliance with treatment assignment. In this article, we establish the nonparametric identification of the complier average direct and spillover effects in two-stage randomized experiments with interference and noncompliance. In particular, we consider the spillover effect of the treatment assignment on the treatment receipt as well as the spillover effect of the treatment receipt on the outcome. We propose consistent estimators and derive their randomization-based variances under the stratified interference assumption. We also prove the exact relationships between the proposed randomization-based estimators and the popular two-stage least squares estimators. The proposed methodology is motivated by and applied to our own randomized evaluation of India's National Health Insurance Program (RSBY), where we find some evidence of spillover effects. The proposed methods are implemented via an open-source software package. for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement. [ABSTRACT FROM AUTHOR]