In the woodlands of sub-Saharan Africa, a coevolutionary arms race is underway. African cuckoos lay their eggs in fork-tailed drongo nests, and those drongos are left raising the cuckoo chick. But drongos are not defenceless. Each female has her own egg ‘signature’ – made up of a combination of eggshell colours and marking patterns – and can reject a cuckoo egg if its appearance differs from this signature. Despite cuckoos producing near-perfect mimetic forgeries, they are not able to preferentially parasitise drongo clutches that match their egg type. This means that, on average, their forgery will be a poor match for the drongo nest it is laid in. We show that due to this, only 6% of cuckoo eggs are predicted to escape rejection by drongos. This demonstrates that even when mimicry is near-perfect, the mimic may not have the upper hand in the arms race.