In general a state will tax you if you 1) live in that state; 2) earn money in that that; or 3) make money from property in that state.
So it appears that you file and pay in both MI/MN but your MI will give you a credit for the tax paid to the other state.
See the MI 1040 instructions page 10, for line 18.
https://www.michigan.gov/taxes/-/media/Project/Websites/taxes/Forms/2023/2023-IIT-Forms/BOOK_MI-1040...
I haven't reviewed the MN instructions, so you should double check those.
I would do the MN return first in TT, then do your resident state last. When doing the MN return you will (if it is like other states I've seen) indicate for each piece of income (wages, interest, dividends, capital gain, etc.) whether it came from MN sources. If you didn't live in MN only the rental property would be MN sourced.
Then do the MI return. They carefully review the forms to see how how it works. If you don't have the right numbers on one of the returns redo the interview. I would be looking on the MN return to see that you were only taxed on the rental sale (and any rental income prior to the sale). On the MI return I would be looking to see if the line 18 numbers look like you're getting the credit for any MN tax liability.
Do note that I am not a MN/MI tax expert. I just googled and quickly read the instructions. You should do you own research and/or consult a tax professional to make a decision as to how to proceeded.
The sale of rental property is somewhat complicated because of your prior depreciation and any allocation you make to non-real estate items included in the sale (appliances, furnishings, etc.). The above doesn't get into any of that.
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