We present LAReQA, a challenging new benchmark for language-agnostic answer retrieval from a multilingual candidate pool. Unlike previous cross-lingual tasks, LAReQA tests for “strong” cross-lingual alignment, requiring semantically related cross-language pairs to be closer in representation space than unrelated same-language pairs. This level of alignment is important for the practical task of cross-lingual information retrieval. Building on multilingual BERT (mBERT), we study different strategies for achieving strong alignment. We find that augmenting training data via machine translation is effective, and improves significantly over using mBERT out-of-the-box. Interestingly, model performance on zero-shot variants of our task that only target “weak” alignment is not predictive of performance on LAReQA. This finding underscores our claim that language-agnostic retrieval is a substantively new kind of cross-lingual evaluation, and suggests that measuring both weak and strong alignment will be important for improving cross-lingual systems going forward. We release our dataset and evaluation code at https://github.com/google-research-datasets/lareqa.