One solution is to have a single developer make all the changes across products. This makes sense if all the products use the same infrastructure, or all have high standards of documentation and devops. E.g. it doesn't work well if different products have different build processes that all require manual intervention. When we reduce barriers to entry this way, "ownership" of individual products might then be meaningful only for code reviews and maintaining long term health of each code base.
The only other solution is to have developers on each of the products all working together to integrate early. If your sprints are two weeks long, each developer gets one week to build an initial implementation, and must be available to integrate immediately on the first day of the second week. Everyone should expect developers to refactor their initial implementations afterwards.