While filling in a job application, you often have an option “contact the referees after the interview”. Don’t be scared to use it, most of the prospective employers only contact the referees after the interview. If you were shortlisted for an interview, you already have good chances to get the job. After the interview the potential employers tend to rely more on a good in person experience, than on a reference from an unknown person with a potential conflict of interests. Just be honest why you don’t want to ask for a reference from your previous supervisor or PI.
