I had four plays running at once

Hi! Could you help with English grammar?

**The fact that I had four plays running at once brought me great notoriety.**
I wonder whether 'running' in the above sentence is an object complement or not.
If so, I don't understand it. Because I know that 'have' the causative verb doesn't take 'root verb' as an object complement.

Is 'running' the present participle? Does 'running' modify the noun, plays and does 'have' mean 'possession'?
What does 'running' act like in the sentence?

Thank you for reading.

It sounds like the context is that some playwright or whatever is recounting a particularly successful period in his career.

When a play is giving performances to the public (in other words, they're not just rehearsing) the play is said to be "running", as in:

"Blue Man Group's show is currently running at the Astor Place Theater."
"'Cats' is one of Broadway's longest-running shows."

So your sentence means that the speaker had 4 different plays of his giving performances during the same time period.