Sentiment analysis aims to systematically study affective states and subjective information in digital text through computational methods. Aspect Sentiment Triplet Extraction (ASTE), a subtask of sentiment analysis, aims to extract aspect term, sentiment and opinion term triplets from sentences. However, some ASTE’s extracted triplets are not self-contained, as they reflect the sentence’s sentiment toward the aspect term, not the sentiment between the aspect and opinion terms. These triplets are not only unhelpful to people, but can also be detrimental to downstream tasks. In this paper, we introduce a more nuanced task, Aspect–Sentiment–Opinion Triplet Extraction (ASOTE), which also extracts aspect term, sentiment and opinion term triplets. However, the sentiment in a triplet extracted with ASOTE is the sentiment of the aspect term and opinion term pair. We build four datasets for ASOTE. A Position-aware BERT-based Framework (PBF) is proposed to address ASOTE. PBF first extracts aspect terms from sentences. For each extracted aspect term, PBF generates an aspect term-specific sentence representation, considering the aspect term’s position. It then extracts associated opinion terms and predicts the sentiments of the aspect–opinion term pairs based on the representation. In the experiments on the four datasets, PBF has set a benchmark performance on the novel ASOTE task.