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If your spouse earned income and paid taxes to Missouri, she would file a Non-Resident Missouri return to get her tax refunded. She would report only her income and taxes paid there.
It is best to prepare this return first, before preparing your Joint Illinois Resident return. In Illinois, you report both of your incomes, and any amount of tax she paid to Missouri, which Illinois should give her credit for.
In the Illinois interview, it should ask if 'taxes were paid to another state'. If this was done and Illinois says there is an 'underpayment', it could be that Illinois taxes are higher than Missouri taxes on the same amount of income.
All of your military income is excluded from Illinois tax.
Here's more info:
https://ttlc.intuit.com/replies/3302052
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