Soft robotics technology has been proposed for a number of applications that involve human–robot interaction. It is commonly presumed that soft robots are perceived as more natural, and thus more appealing, than rigid robots, an assumption that has not hitherto been tested or validated. This study investigates human perception of and physical interaction with soft robots as compared with rigid robots. Using a mixed-methods approach, we conducted an observational study to explore whether soft robots are perceived as more natural, and what types of interactions soft robots encourage. In a between-subjects study, participants interacted with a soft robotic tentacle or a rigid robot of a similar shape. The interactions were video recorded, and data was also obtained from questionnaires (Nvideo = 123, Nquest = 94). Despite their drastically different appearances and materials, we found no significant differences in how appealing or natural the robots were rated to be. Appeal was positively associated with perceived naturalness in all cases, however we observed a wide variation in how participants define “natural”. Although participants showed no clear preference, qualitative analysis of video data indicated that soft robots and rigid robots elicit different interaction patterns and behaviors. The findings highlight the key role of physical embodiment and materiality in human–robot interaction, and challenge existing assumptions about what makes robots appear natural.