My wife and I live in MA, she works solely in CT, and I work solely in MA. Her employer only withheld MA state income taxes. How do we file without being double taxed?

Details: we file federal and state jointly (even state because she gets a MA W2 since her employer withholds MA income taxes instead of CT). According to MA website, the employer must withhold and pay taxes to the “work state” (CT). According to CT website, she must file a 1040NR/PY return and file married filing separately. When we go to do this, she owes a significant amount to CT because there’s no where to show taxes were paid to MA. When in reality she should be getting some sort of credit, or more accurately put... taxes should be paid to CT and if MA taxes are higher then MA should give a credit and receive the difference. 

How do we know what portion of her MA withholdings were paid to CT by her employer? How do we file the 1040NR/PY for CT without being double taxed?