We tackle the problem of visual search under resource constraints. Existing systems use the same embedding model to compute representations (embeddings) for the query and gallery images. Such systems inherently face a hard accuracy-efficiency trade-off: the embedding model needs to be large enough to ensure high accuracy, yet small enough to enable query-embedding computation on resource-constrained platforms. This trade-off could be mitigated if gallery embeddings are generated from a large model and query embeddings are extracted using a compact model. The key to building such a system is to ensure representation compatibility between the query and gallery models. In this paper, we address two forms of compatibility: One enforced by modifying the parameters of each model that computes the embeddings. The other by modifying the architectures that compute the embeddings, leading to compatibility-aware neural architecture search (CMP-NAS). We test CMP-NAS on challenging retrieval tasks for fashion images (DeepFashion2), and face images (IJB-C). Compared to ordinary (homogeneous) visual search using the largest embedding model (paragon), CMP-NAS achieves 80-fold and 23-fold cost reduction while maintaining accuracy within 0.3% and 1.6% of the paragon on DeepFashion2 and IJB-C respectively.