Twitter has emerged as a global hub for engaging in online conversations and as a research corpus for various disciplines that have recognized the significance of its user-generated content. Argument mining is an important analytical task for processing and understanding online discourse. Specifically, it aims to identify the structural elements of arguments, denoted as information and inference. These elements, however, are not static and may require context within the conversation they are in, yet there is a lack of data and annotation frameworks addressing this dynamic aspect on Twitter. We contribute TACO, the first dataset of Twitter Arguments utilizing 1,814 tweets covering 200 entire COnversations spanning six heterogeneous topics annotated with an agreement of 0.718 Krippendorff’s α among six experts. Second, we provide our annotation framework, incorporating definitions from the Cambridge Dictionary, to define and identify argument components on Twitter. Our transformer-based classifier achieves an 85.06% macro F1 baseline score in detecting arguments. Moreover, our data reveals that Twitter users tend to engage in discussions involving informed inferences and information. TACO serves multiple purposes, such as training tweet classifiers to manage tweets based on inference and information elements, while also providing valuable insights into the conversational reply patterns of tweets.