In Liuzhi Special Zone, as far as I know, it is necessary to provide relevant materials to apply for household registration certificate, including ID card, household registration book and identity certificate. Identity cards alone are not enough. If your girlfriend is registered with your local parents, you may need your parents' consent and support to handle it smoothly.
In addition, the household registration certificate is only a document to prove the personal household registration information, and it does not have much influence on your love relationship itself. If you want to be together, you can solve practical problems through communication and find suitable solutions to promote the relationship.