I'm surprised that people are confused. Is there a regional difference in the way this would be phrased? In my (US) experience, "engaged with" unambiguously means working together, while "engaged to" unambiguously means getting married.

Indeed, at least from my American Midwest perspective, I agree it's all in the conjunction, with no ambiguity at all.

I didn't notice any ambiguity until I read the replies

I immediately thought engaged -> married. I didn't even pay attention to the with/to conjunction. Anything "engaged" means marriage to me unless presented as something that is not a person. I get it now, and it makes sense, but I have a feeling there are a lot of people who would think, programmer finds female programmer that is excellent at what she does and wants to "engage" her.

Phrases like "I tried to call, but they were engaged." must really confuse you then.

There are a lot of things in the world that are engaging, many would say far more engaging, than just the prospect of marriage.

The same here, in the UK.

I understood what he meant, but it still made me double-take.