You’re kidding, right? Of course we would. (Unless it’s chickenshit business, like Indonesia and Oz getting into a coast guard squabble over migrant boats.)
When you co-produce weapons, you send high-ranking technical advisors and expensive kit that needs guarding. This is why e.g. deploying a ballistic missile radar isn’t just plopping down a radar. It involves deploying troops around it. We have defence treaties with countries whom we don’t trust enough to co-produce with. Co-production is a big deal.