The most common advice for operating without authority is to build relationships and align incentives. That is true and almost useless as practical guidance. Here is what it actually looks like.
You align without authority by making the cost of misalignment visible earlier than anyone else does. When team A's plan conflicts with team B's timeline, most people notice when the collision happens. Your job is to surface it before either team commits to something they cannot deliver. Whoever surfaces the problem first owns the solution, and that is where influence lives.
At Salesforce, I ran a portfolio where I had no authority over any of the eight teams I depended on. What I had was the dependency map, which meant I was the first person to see conflicts. I made that map visible to everyone, which meant the teams started coming to me before conflicts hardened into commitments. That is a position of influence. I did not earn it through relationships. I earned it through information.
The other thing that works is being consistent about what you will and will not absorb. If you take on problems that are not yours to solve, you train teams to bring you their problems. If you are clear about what belongs to the program versus what belongs to a team, you train them to solve the right things themselves.
Influence without authority is mostly about information and consistency. The relationships matter, but they follow from those two things, not the other way around.