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Automatic evaluation of speech-to-text translation (ST) systems is typically performed by comparing translation hypotheses with one or more reference translations. While effective to some extent, this approach inherits the limitation of reference-based evaluation that ignores valuable information from the source input. In machine translation (MT), recent progress has shown that neural metrics incorporating the source text achieve stronger correlation with human judgments. Extending this idea to ST, however, is not trivial because the source is audio rather than text, and reliable transcripts or alignments between source and references are often unavailable. In this work, we conduct the first systematic study of source-aware metrics for ST, with a particular focus on real-world operating conditions where source transcripts are not available. We explore two complementary strategies for generating textual proxies of the input audio, automatic speech recognition (ASR) transcripts, and back-translations of the reference translation, and introduce a novel two-step cross-lingual re-segmentation algorithm to address the alignment mismatch between synthetic sources and reference translations. Our experiments, carried out on two ST benchmarks covering 79 language pairs and six ST systems with diverse architectures and performance levels, show that ASR transcripts constitute a more reliable synthetic source than back-translations when word error rate is below 20%, while back-translations always represent a computationally cheaper but still effective alternative. Furthermore, our cross-lingual re-segmentation algorithm enables robust use of source-aware MT metrics in ST evaluation, paving the way toward more accurate and principled evaluation methodologies for speech translation.