Hi OP, I was in a similar but slightly different situation, so I understand how hard and confusing this must be. In my case, he wasn't married, but his family were keen for him to meet and marry someone from their own culture. He was quite liberal, and had other ideas.
I spent years in the relationship, waiting. He became a firm part of my family, my parents treated him like their own son. My nephews and nieces called him 'uncle'. I never met his family. It was brutal.
We couldn't live together, I wasn't allowed to visit him (he lived with his family, despite being late 20s). I knew his parents were continually introducing him to women that they deemed suitable. It was brutal.
He told me that they knew I existed, but to this day I don't know if they truly did. I got tired of waiting around and realised that if I had to take control of my future. It hurt like hell, it took a long time to get over him - but I honestly don't regret my choice at all. I'm now happily married with a DS and I honestly feel that had I not made the decision for myself, I would possibly still be there waiting to stop being a secret.
After breaking up he did regret it, I received daily emails and he tried to spend the next 2 years persuading me otherwise. I mostly ignored it. There was a moment where I always almost persuaded, but when faced with an ultimatum, even then he couldn't do it. To me that was very telling.
I spent years making excuses for him. We both made excuses for his inability to make that jump - cultural differences became the perfect excuse. In the end it was simple, just couldn't do it.