In ISAs where this problem exists, i.e. in x86-64 where the carry flag is updated or not updated regardless of what happens with the other flags, the flags register is split into 2 separate registers, which are renamed independently.

The 2 parts of the flags register are reunited only for the purpose of saving or restoring from the main memory. (Actually the x86-64 flags register has 3 independent parts, the carry flag, the other status flags, and a set of configuration flags, which are not modified by most instructions.)