Simplify value selection
You have four cases and each of them behaves differently, so you actually need to have four different branches. But you can simplify your code by extracting the checks into variables and using nested ifs:

public string CallKey
{
    get
    {
        bool callKeyEmpty = string.IsNullOrEmpty(CALLKEY);
        bool asCallKeyEmpty = string.IsNullOrEmpty(AS_CALL_KEY);

        if (callKeyEmpty) 
        {
            if (asCallKeyEmpty)
                return "";
            else
                return AS_CALL_KEY.Trim();
        }
        else
        {
            if (asCallKeyEmpty)
                return CALLKEY.Trim();
            else
            {
                if (AS_CALL_KEY.Trim() != CALLKEY.Trim())
                    _error = true;
                return AS_CALL_KEY.Trim();
            }
        }
    }
}


As an added benefit, you can now remove the “this never happens” part.