In this paper, we study Condorcet domains, sets of linear orders from which majority ranking produces a linear order. We introduce a new class of Condorcet domains, called coherent domains, which is natural from both a voting theoretic and combinatorial perspective. After studying the properties of these domains we introduce set-alternating schemes. This is a method for constructing well-behaved coherent domains. Using this we show that, for sufficiently large numbers of alternatives n, there are coherent domains of size more than 2.1973n. This improves the best existing asymptotic lower bounds for the size of the largest general Condorcet domains.