There are 4 other states and none of them have agreements with Illinois. I paid state taxes to the 4 other states, but do I need to pay tax on that to Illinois as well? I am being told I can't efile because I am not claiming all my income as IL income.
Yes, you do. Your resident state taxes all of your income regardless of where you earn it. As you mention, when the two states have a reciprocal agreement, the nonresident state agrees not to collect tax on your income. However, when this agreement is not in place, you are required to file a nonresident return for all of the income earned individually in other states, and a resident return claiming all of the income to your home state (IL). However, to reduce the effect of double-taxation, Illinois, your resident state, will allow you to receive a tax credit for the tax you are paying to the other state (and, perhaps locality as well) on the amount of income you earned in that state. Here's how the credit works.
You live in Illinois, and work in Missouri and Nebraska (neither of which has a reciprocal agreement). You earn 5,000 each in Missouri and Nebraska. Missouri taxes you 300 on that income, and Nebraska taxes you 150. Illinois tax on the 10,000 is 400. Illinois will give you a $200 credit for the tax you pay to Missouri (of the $300 you had to pay), and $150 for the tax you paid to Nebraska. You would still owe Illinois $50 for the amount not covered by the Nebraska tax credit.
So, yes, you do need to report all of the income to Illinois, but this credit will assist so that the effects of double-taxation is greatly reduced.
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Our company has an employee who is a resident of Illinois and will work remotely from his house in IL 3 days per week. He will attend meetings and work in Missouri 2 days per week. Does our company need to collect non-resident income taxes for Missouri as well as resident income taxes for IL? Another twist, our company is located in Massachusetts, the employee is working at a client's facility 2 days per week in MO. Of course, he is working full time from home right now due to the pandemic.