Abstract We investigate the use of two or more linked lists, for both population size estimation and the relationship between variables appearing on all or only some lists. This relationship is usually not fully known because some individuals appear in only some lists, and some are not in any list. These two problems have been solved simultaneously using the EM algorithm. We extend this approach to estimate the size of the indigenous Māori population in New Zealand, leading to several innovations: (1) the approach is extended to four lists (including the population census), where the reporting of Māori status differs between registers; (2) some individuals in one or more lists have missing ethnicity, and we adapt the approach to handle this additional missingness; (3) some lists cover subsets of the population by design. We discuss under which assumptions such structural undercoverage can be ignored and provide a general result; (4) we treat the Māori indicator in each list as a variable measured with error, and embed a latent class model in the multiple system estimation to estimate the population size of a latent variable, interpreted as the true Māori status. Finally, we discuss estimating the Māori population size from administrative data only. Supplementary materials for our article are available online.