Is the picture of the real robot showing the robot at the joint values highlighted on your teach pendant? (all joints at 0 deg except for A2 at -90 and A3 at +90)
I've never seen a KUKA robot using a KUKA KRC controller that has a different home position from the one shown in RoboDK. I assume you have a KRC2 or a KRC4 controller.
You may need to follow the mastering procedure to recover the home position of the real robot. This can happen if your robot has been turned off for a long time. These robots usually have a battery to remember the pulse count so you may need to replace that battery as well if you don't want to do this mastering procedure every time you turn off your robot.
I was about to say the same as Albert, that's a very weird home position.