What affiliation do I choose if my scientific work is unrelated to my job?

There are two main paths here:

• You can list yourself as an independent researcher, which is appropriate if you are not doing this "on the clock" at your current position and use no company resources.
• You can list yourself under your current employer, which is appropriate if you are doing this using work time and/or company resources, even if the work is not related to your position's responsibilities.

If you are working at a company that participates in research (even if in another field), they probably have policies that apply (possibly including pre-submission review), and you should have a discussion with your supervisor and/or other folks who are engaged in research there. A clause in your employment agreement might even mean that you cannot be an independent researcher.

If, on the other hand, you are working for a completely non-research organization (e.g., a paper company), then you probably should be doing this as an independent researcher and keeping everything off the clock.

Finally, there is a somewhat more complex third possibility to also consider: if these collaborations are part of a long-term relationship with a university, your collaborators might be able to get you some sort of "visiting researcher" or "associated researcher" affiliation. These "positions" have no money associated with them and are really just there to simplify the logistics of collaboration, so they are often pretty simple to arrange, and could actually be the most appropriate description of your position with regards to your research.

As someone who developed a publication policy and approval process for the company I work for, I suggest that you first make sure that you really have no publication policy by asking your manager, the marketing/communications dept, or legal (in this order for ease of life).

A few points to consider

• Your biggest (potential) problem is actually breach of contract with regard to intellectual property. You first need to make sure that you're not barred from publishing by your employment agreement. Not that you should avoid publishing if that's the case, but then I would definitely not talk to your company about it nor mention them in any way.

• Making your current company an affiliation means claiming that the contents is the company's position and/or claims. Even if the association with the company is not of full strength - it's definitely there, i.e. you're committing them to something. So do ask them before putting their name on.

• I agree with @jakebeal's criterion: If you used company resources to get this research done / published, you should offer the company affiliation. If you didn't, you should probably be listed as independent.

• If the research started before you left academia, you could list the academic institute still, and add a footnote saying "Research conducted/conducted in part while XYZ at ABC institute" or something to that effect.