Zero-shot translation aims to translate between language pairs not seen during training in Multilingual Machine Translation (MMT) and is largely considered an open problem. A common, albeit resource-consuming, solution is to add as many related translation directions as possible to the training corpus. In this paper, we show that for an English-centric model, surprisingly large zero-shot improvements can be achieved by simply fine-tuning with a very small amount of multi-parallel data. For example, on the EC30 dataset, we obtain up to +21.7 ChrF non-English overall improvements (870 directions) by using only 100 multi-parallel samples while preserving English-centric translation quality. When investigating the size effect of fine-tuning data and its transfer capabilities, we found that already a small, randomly sampled set of fine-tuning directions is sufficient to achieve comparable improvements. The resulting non-English performance is close to the complete translation upper bound. Even in a minimal setting -- fine-tuning with only one single sample -- the well-known off-target issue is almost completely resolved, explaining parts -- but not all -- of the observed improvements in translation quality.
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