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How adult stem cells maintain self-renewing tissues is commonly assessed by analysing clonal data from in vivo cell lineage-tracing assays. To identify strategies of stem cell self-renewal requires that different models of stem cell fate choice predict sufficiently different clonal statistics. Here, we show that models of cell fate choice can, in homeostatic tissues, be categorized by exactly two ‘universality classes’, whereby models of the same class predict, under asymptotic conditions, the same clonal statistics. Those classes relate to generalizations of the canonical asymmetric vs. symmetric stem cell self-renewal strategies and are distinguished by a conservation law. This poses both challenges and opportunities to identify stem cell self-renewal strategies: while under asymptotic conditions, self-renewal models of the same universality class cannot be distinguished by clonal data only, models of different classes can be distinguished by simple means.