Putting these together seem inevitable, since they are both too extreme in practice. If we are talking about deontology, it becomes next to impossible in practice. If a murderer asks you about his next victims whereabouts, do you lie? And the extreme situation for consequentialism would be allowing gladiator battles, since the overall pleasure would outweigh the suffering of the gladiators.
There are situations where you lean towards deontology and situations where you lean towards consequentialism. What if we make the synthesis situationwise? The theory itself won't have problems with situation X and Y, since it will adopt to these situations? The problem with consequentialism and deontology are these situations, and lets start with consequentialism:
It's inevitable that someone will die in the current going of things, the only thing you can change is that will X (x<y) die or Y (y>x) amount of humans. All of them are in the situation, and can't leave the situation before it is over. The most used is the train example: There are five humans on the track where you are going, and one on the right track. The brakes won't be able to stop the train in time, and either the five die or the one dies. Will you turn the train to kill the one, or stay on the track killing the one?
Now lets see the argument for deontology:
It's not inevitable that y (y>x) people die in situation, if you can stop it by killing innocent life of x. Most used example is the traffic accident. There are five injured people who will die if they don't get the organ transplantation. There are five different failures of organs in these five people. So if the doctor were to kill the one innocent man on the hospital, he would be able to save these five.
These are not of course definitive arguments for their side, but are pretty compelling arguments to make someone doubt their position. Hence the synthesis would be required.