State Tax Witholdings

I live in MI but am working onsite in WI for a client. I opted to rent an apartment in WI close to the site as this was the cheapest option compared to traveling/hotel fees. This was not an issue as my office location was in MI and taxes were being taken out for MI. Now, however, my company has switched my office to WI. I have read that WI and MI have a reciprocity agreement, but now I'm wondering if renting an apartment in WI complicates the tax issue. Am I legally required to declare the WI apartment as my residence since I am staying there for work? Can I leave my residence as my MI address and continue to have taxes taken out for MI only? 

 

I am not sure what the best approach is and would greatly appreciate any advice!