Is "common life" in common usage?
Solution 1:
I don't think 'common life' is anything like a standard phrase or collocation, especially in the sense you're using it.
For me, the easiest accessible sense of 'common life' would probably be the same as something like 'the average life of an average person'.
Nevertheless, I don't see any problem with using it as you seem to want to, as something like the intersection between two people's lives. For example, you could say:
"We sleep in the same bed. We drink the same coffee. We have the same friends, the same fears, and the same damn foibles! We share a common life!"