If either device can write to the register and both keep a copy, then if there's a difference, you need to figure out which one to use, which may be difficult. One way around it is to store the controlling copy of the register in only one location, preferably the one that needs that data most. For example, if I have a register that can be written by the operator via the HMI, but can also be overwritten by the PLC, I have the HMI always displaying the PLC register, and also writing to it. If the PLC overwrites, you still see the up-to-date value on the HMI.
Both devices can read the register, both can write to it, but you never have to arbitrate which is the "right" value.