Although field-collected recordings typically contain multiple simultaneously vocalizing birds of different species, acoustic species classification in this setting has received little study so far. This work formulates the problem of classifying the set of species present in an audio recording using the multi-instance multi-label (MIML) framework for machine learning, and proposes a MIML bag generator for audio, i.e., an algorithm which transforms an input audio signal into a bag-of-instances representation suitable for use with MIML classifiers. The proposed representation uses a 2D time-frequency segmentation of the audio signal, which can separate bird sounds that overlap in time. Experiments using audio data containing 13 species collected with unattended omnidirectional microphones in the H. J. Andrews Experimental Forest demonstrate that the proposed methods achieve high accuracy (96.1% true positives/negatives). Automated detection of bird species occurrence using MIML has many potential applications, particularly in long-term monitoring of remote sites, species distribution modeling, and conservation planning.