The best open source alternative to Redbooth is Redmine, which is both free and Open Source. If that doesn't suit you, our users have ranked more than 100 alternatives to Redbooth and many of them is open source so hopefully you can find a suitable replacement. Other interesting open source alternatives to Redbooth are Taiga.io (Freemium, Open Source), Restyaboard (Freemium, Open Source), HumHub (Freemium, Open Source) and TaskBoard (Free, Open Source).
A Kanban-inspired app for keeping track of things that need to get done. The goal of TaskBoard is to provide a simple and clean interface to a functional and minimal application for keeping track of tasks. It's not trying to be the next Trello or LeanKit.
ZenTao is an open source project management tool, supporting Scrum. It integrates product management, project management, bug management and QA management, etc., covering all lifecycle of software development.