registration requirements.

Let say a company's having its Corporate office in mumbai(Providing no taxable service from mumbai) and another office in Delhi(Providing taxable service from delhi). Whether the company is required to obtain a fresh registration in mumbai or not ?
And if yes, then receiving any taxable service from an unregistered person(From Mumbai) by company in mumbai is subject to RCM u/s 9(4) of CGST. The ITC of such service can only be adjust from the output tax liability of mumbai and not from output tax liability of delhi. since no output tax liability is arise for the comapny in the state of mumbai. How comapny can claim ITC Occurs in mumbai.?