We develop the mathematical structure of the neutral coalescent with both replication-dependent and replication-independent mutations. This allows us to explain and quantify empirical results that explore differences in genetic diversity in bacterial cultures with different growth rates. We also derive an unbiased and consistent estimator for the replication-independent mutation rate that is based on a comparison of total single nucleotide polymorphism counts for two independent well-mixed cultures with different growth rates. In addition to explaining differences in genetic diversity between well-mixed cultures with different (but constant) growth rates, our coalescent also quantifies the effects of fluctuating growth rates, a situation that can be common in natural populations.