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You will have to file a non-resident return in Wisconsin to pay whatever tax you owe them based upon your Wisconsin earnings.
Then your normal Michigan Resident return will give you a credit against your Michigan tax for any tax you paid to Wisconsin (up to the Michigan tax on the same income), thus eliminating the double tax.
You'll need to attach a copy of your non-resident Wisconsin tax return to your Michigan return to support the credit.
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Hi and welcome to JustAnswer:
Different expert here.
Wisconsin has a reciprocity agreement with Michigan. If you only Wisconsin income is from wages, you are not required to file a Wisconsin return unless you need to recoup Wisconsin tax withheld.
You should advise your employer to withhold Michigan tax, not Wisconsin. If your employer is unwilling or unable to withhold Michigan tax you will need to make estimated tax payments to Michigan.