If you've been in this game long enough, it's very likely you've read Arnold's Encyclopedia of Modern Bodybuilding. In it, Arnold wrote about the various splits he used in the heyday of his training.
One of his most common splits was training chest and back together. He'd just go back and forth between exercises for the chest and back. He did the same for other opposing body parts, too, like biceps and triceps.
He didn't need 78 peer-reviewed studies to tell him that it worked. Through common sense, he settled on a method where he worked an antagonist muscle group and its agonist muscle group on alternating sets.
Arnold's lessons have endured the test of time and experience. Studies have confirmed that pressing strength increases dramatically by working or even statically stretching the antagonist muscles between sets of benching. Additionally, strength experts have used this method very effectively for increasing strength in their athletes for quite some time.
Agonist/antagonist training also ensures that you're doing enough work for both sides of the body so that you build and/or maintain muscular balance. Similarly, you don't want to get too "press heavy" (overdeveloped front delts that contributes to poor posture) and you want to make sure the hamstrings get as much work as the quadriceps.
Another factor in putting this all together is how to go about moving in between exercises. Arnold would often superset between his pushing and pulling movements. However, a rest between movements can prove far more beneficial.
Try doing a set of bench presses, followed by a 2-3 minute rest before proceeding to the pulling movement. Then rest again for 2-3 minutes before going back to benching. This is called alternating sets, and I've found it to be far more beneficial in regards to increasing strength than supersetting the agonist/antagonist movements.
TAKE INSPIRATION FROM THE BIGGEST PIONEER AND UNDISPUTED KING OF BODYBUILDING, YOU WON'T REGRET IT!