In large clonal populations, several clones generally compete which results in complex evolutionary and ecological dynamics: experiments show successive selective sweeps of favorable mutations as well as long-term coexistence of multiple clonal strains. The mechanisms underlying either coexistence or fixation of several competing strains have rarely been studied altogether. Conditions for coexistence has mostly been studied by population and community ecology, while rates of invasion and fixation have mostly been studied by population genetics. In order to provide a global understanding of the complexity of the dynamics observed in large clonal populations, we develop a stochastic model where three clones compete. Competitive interactions can be intransitive and we suppose that strains enter the population via mutations or rare immigrations. We first describe all possible final states of the population, including stable coexistence of two or three strains, or the fixation of a single strain. Second, we give estimate of the invasion and fixation times of a favorable mutant (or immigrants) entering the population in a single copy. We especially show that invasion and fixation can be slower or faster when considering complex competitive interactions. Third, we explore the parameter space assuming prior distributions of reproduction, death and competitive rates and we estimate the likeliness of the possible dynamics. We especially show that when mutations can affect competitive interactions, even slightly, stable coexistence is likely. We discuss our results in the context of the evolutionary dynamics of large clonal populations.