I am considering a move to another institution, two years out from tenure. I do have a strong case, but the department has changed in ways that may mean better prospects elsewhere. I am not 100% set on leaving; rather I will be looking. I do think my current department would frown on my looking or actively soliciting a position. Also, my field is small; word will likely get around. How do people do this, correctly and strategically?
It is difficult. Job application lists are supposed to be confidential, and you can mention in your cover letter that you prefer your institution not be notified of your candidacy. In some countries, your privacy is much better protected as an applicant. For example in Denmark, faculty applicants don't give talks and there's no public record of them even having an on-campus interview. Consider applying strategically depending on who is likely on the hiring panel at what types of schools, and how likely that is to come back. Last, it's very normal for faculty to apply to other positions, and even if discovered, may not affect your tenure case or your relationship with colleagues. Good luck.