The phrase "barge in on" often has a negative connotation, implying that someone is interrupting or intruding without invitation. Some synonyms for this phrase could include "intrude on," "burst in on," "walk in on," "crash in on," or "break in on." These phrases all imply a lack of boundaries or consideration for others. However, there are also ways to enter a conversation or situation without intruding - synonyms for this could include "join in," "participate," "contribute to," or "engage with." It is important to be mindful of others' boundaries and respect their space, rather than forcing oneself into someone else's conversation or situation.