I am a PhD student in the social sciences and I already have a first advisor.
A few years ago, eyeing the possibility of a PhD, I had informally asked a professor of mine if she would be possibly interested/have the capacity to be my advisor. I had already worked with her and liked the collaboration very much. She is very competent on the data analysis techniques I will use, knows my specific subfield really well, and has a good network. She has invited me twice to discuss my project ideas informally in her team meetings with her other PhD and postdocs, despite me not being an official member of the team. All in all, she has been really encouraging and supportive.
However, she has some ongoing conflicts with different members of the department (I do not know exactly why, who initiated them, etc.). I am not sure about the relationship between her and my first advisor. For sure, the first thing I will do is talk with my first advisor about it. I would like to avoid any forms of conflict in my committee. Also, the techniques she is an expert on might have a relatively minor/subordinate role in the whole architecture of the dissertation.
In case my first advisor expresses preferences for other professors, how should I deal with turning down her offer to supervise my work? Have I behaved unethically by expressing interest, profiting from her feedback, and now having to retract, and if yes, how can I make up for it?
I would especially like to further the collaboration in the future, even if it is not in the form of advisor-PhD student role. I am thinking about telling her that my first advisors suggested other co-advisors and that the project will use, for the most part, a different methodology, so other advisors might be a better fit. I am also thinking of asking her if we can further collaborate and suggest co-authoring a paper about that specific subsection of the PhD dissertation. However, I will also have to ask my first advisor if she agrees to that.
Do you have any thoughts on how to approach this situation and how to best communicate with this professor?