What do I select as my residence / status if I am a non-resident but have "Residential Ties" in Montreal meaning I will need to submit tax information?

If you are a non-resident in Canada but you have to submit a Quebec return, then you will need to do two returns in TurboTax. On one you would say that you are non-resident and submit that to the Canada Revenue Agency (CRA). On the second one you would need to choose Quebec as your province, so that you can access the Quebec forms to send to Revenu Québec. 

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