This may be possible through GitHub Projects. Projects exist per the organization, not a particular repo. You can also link to issues and pull requests from repos of other organizations by entering user/repo#issue
in the project's task description.
In the case of your scenario, suppose your team shares a GitHub organization. You can create project, then add items that reference your issues and pull requests, which exist within the open source project's repo. You can configure and organize your project's view(s) to display the information you want to show. Since this merely links to the issues/PRs from the open source repo, the original stays intact, and how they are displayed in your project stay in sync.