Given a set of results from range minimum queries (RMQs), our task is to construct a sequence that is consistent with the results of the queries. We study two types of RMQs: a value-based RMQ returns the minimum value and an index-based RMQ returns the index of the minimum. While the value-based version has been discussed informally in the context of competitive programming, the index-based version appears to be unexplored. In this paper, we provide a survey and unified analysis of the value-based version, and we propose efficient algorithms for the index-based version. These include algorithms for computing the lexicographically smallest consistent sequence and permutation, as well as an enumeration algorithm that outputs all consistent permutations with constant delay.