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A quote tweet (also known as a "retweet with comment") is a reply on Twitter to an original tweet that displays the entirety of the original in a reply-context below the reply text.
Quote tweet is available in the Twitter UI as an option inside their "retweet" button.
With reply text commentary:
If someone posts a quote tweet in response to a POSSE tweet copy of one of your posts, it makes sense to backfeed it to your original like any other @-reply tweet.
Bridgy does support backfeeding quoted tweets. Add your examples / opinions to the feature request:
Articles about quote tweets and similar constructs: