See Pondini's TM FAQs, for details. AFAIK, the two different OS have OS-specific TM apps running and I'm not surprised that they're not forward or backward compatible. As for better solutions, bootable backups, IMO, are superior. For those I recommend Carbon Copy Cloner.
So, lets say that Tima Machine apps are OS-specific. Then if the iMac is updated to Mavericks it the files in the backups for both machines can be read by both machines? I wonder if anyone has tried this and can confirm that if the Time Machine app version is the same for two seperate backups of two separate computers, then both computers will be able to read the files in both backups?
If Time Machine apps are no forward and backward compatible, that is a deal breaker for me. I am surprised Apple has not thought this through more thoroughly. Developing software that assumes that consumers are using only one operating system seems a little short sighted.
Thanks for the tip about Carbon Copy Cloner, it is worth looking into.
Rather than install a buggy Mavericks on another machine, I downloaded Carbon Copy Cloner to give it a try. So far it has the same issue as Time Machine. The MacBook Air with Mavericks backup, made using Carbon Copy Cloner, does not display files on the iMac with OS10.7.
However, if files are dragged and dropped from the MacBook Air with Mavericks, to the La Cie drive, then they are visible when the La Cie drive is connected to the iMac with 10.7.
My issue seems to have something to do with the backup software, both Time Machine and Carbon Copy Cloner.