GST REGISTRATION

We are a construction company based in Haryana and registered under Haryana GST. We are tendering for a project in Rajasthan for the construction of a factory building. Our query pertains to GST compliance requirements for this project: Do we need to obtain a separate GST registration in Rajasthan for this project, or can we operate using our Haryana GST registration? If we can work under our Haryana GST registration, how should we handle invoicing and tax liabilities (e.g., should we charge IGST or any other tax)? Would establishing a temporary site office or using other facilities in Rajasthan necessitate mandatory GST registration in Rajasthan?