No. If you choose to file separately, since you don't live in (aren't domiciled) in Wisconsin, you will each only report your own income on your state tax returns. The marital property law (e.g. community property) applies only while both you and your spouse are domiciled in Wisconsin.
No, your spouse will not file Form M1RCR. According to the Minnesota Department of Revenue:
To be eligible for this credit, all of these must apply:
• You were a full-year or part-year Minnesota resident in 2023
• You paid 2023 state income tax to both Minnesota and Wisconsin on the same income
• You were a Minnesota resident when both states taxed the same income
For additional information, see the following publication by the Wisconsin Department of Revenue: Tax Information for Married Persons Filing Separate Returns
Also see the Minnesota DOR Taxes Paid to Another State Credit
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