Genetic exchange in microbes and other facultative sexuals can be rare enough that evolution is almost entirely asexual and populations almost clonal. But the benefits of genetic exchange depend crucially on the diversity of genotypes in a population. How very rare recombination together with the accumulation of new mutations shapes the diversity of large populations and gives rise to faster adaptation is still poorly understood. This paper analyzes a particularly simple model: organisms with two asexual chromosomes that can reassort during rare matings that occur at a rate r. The speed of adaptation for large population sizes, N, is found to depend on the ratio ∼ log(Nr)/log(N). For larger populations, the r needed to yield the same speed deceases as a power of N. Remarkably, the population undergoes spontaneous oscillations alternating between phases when the fittest individuals are created by mutation and when they are created by reassortment, which—in contrast to conventional regimes—decreases the diversity. Between the two phases, the mean fitness jumps rapidly. The oscillatory dynamics and the strong fluctuations this induces have implications for the diversity and coalescent statistics. The results are potentially applicable to large microbial populations, especially viruses that have a small number of chromosomes. Some of the key features may be more broadly applicable for large populations with other types of rare genetic exchange.