LLM-based chatbots are now being specifically designed to facilitate social companionship, even romantic relationships, incorporating features that parallel human relationship dynamics. This has led a subset of users to form romantic relationships with chatbots. Understanding which interpersonal characteristics drive individuals to form intense, emotional bonds with chatbots is crucial for comprehending the potential psychological and societal impacts of romantic human-chatbot relationships. This mixed-methods study investigates psychological predictors of relationship intensity among individuals currently in romantic relationships with chatbots. Romantic and sexual fantasy, promising constructs not previously investiagted in this context, are examined alongside previously discussed factors (loneliness, anthropomorphism, attachment orientation, and sexual sensation seeking). In Study 1, quantitative data from individuals with chatbot partners (N=92) showed that romantic fantasy explained the most variance in relationship intensity, with additional contributions from anthropomorphism and avoidant attachment. Contrary to expectations, the other predictors, including loneliness, did not significantly predict intensity. In Study 2, 15 qualitative interviews illuminated how users employ romantic fantasy to enhance their relationships, describing active fantasy use to shape interactions and a desire for their chatbot to feel as human as possible. This study provides the first quantitative sample of this under-researched population, explaining who might form more intense romantic relationships with chatbots.