What can you do to rebuild trust with a team after a project failure?

First, create a culture where people feel safe to speak up, and are empowered to intervene before a project gets to the point of failure. There will always be signs and points in time along the path where, if noticed and acted on, we can course correct and avoid failure. The issue is, most people are afraid to speak up due to the fear of retaliation, being dismissed, or ignored. Second, when failure happens, fail fast, acknowledge it quickly, celebrate and learn from it, be transparent with the data and own it. Then turn it into a positive experience, even if the outcome missed the mark. Don't let it become the 'elephant in the room' that bleeds money, resources and people. Be accountable, don't blame, grow, change, do better next time.

It's important to foster an environment where people feel comfortable speaking up and expressing their concerns or frustrations. By openly communicating what went wrong, why it happened, and what you've learned from it, you show your team that you are accountable and willing to take responsibility for your actions. Debriefing and check-ins are great practices to ensure that everyone involved has the opportunity to share their thoughts and feelings. Remember, rebuilding trust takes time and consistent effort.

We sometimes forget to acknowledge other people's perspective on our failure as a leader. I think simply asking your people for an honest opinion on your own perspective of your failure...well...it will open the floor for a new angle or a new take on something else that failed that you might have overlooked.

Agree with Toni's point on the culture where the people feel safe to speak up. To gain trust, admit mistakes openly, no excuses, document the learning, show your accountability with respect and integrity

I've seen a few others mention fostering a culture where the team feels safe and encouraged to provide feedback, and I absolutely echo that thought. Open dialogue is great, and that open line of communication is optimized when it's with teams who trust one another. Some of the best after-action reviews I've had have been from teams that were open, honest and respectful when addressing failures.

This is almost true. I do agree that the second principal is apologise but what you actually have to do for apology is to accept responsibility. You are the PM. Everything that’s good or bad in the project is your responsibility so first take responsibility. If people made mistakes, then those mistakes were on your instruction or misinterpretation of your instruction, so it all comes back to you - own it, and you can build trust.

Most of the data and evidence may come from the project's progress reports. A project without a clear, regular and shared progress report is doomed to failure. Make sure that the Project Leader or the Project Manager is accountable for circulating a progress report to all participants (weekly, or twice a month). It will also help to go back to the root causes, if needed. It may sound quite basic, but it works.