I'm assuming you're creating a rule of evolution for your imaginary world and are not trying to modify known evolution in the Real World
The problem with modifying the rules of evolution in the Real World is that Earth's evolutionary pattern is one of competition and the "path of least resistance." The critters that could gather the food, protect themselves, get mates... and do that more efficiently than some other critter... they're the critters who survived to evolve. The idea of giving something away (and giving away your ability to propagate is a big one) simply doesn't come into play in Terrestrial evolution.
But let's ignore that. It's boring.
I am going to make one assumption. You didn't explain, but it seems this question is only valid if there is a bonded pair, a married couple, if you will, and that bonding/marriage/union is desirable as a species (and not just as a governmental, legal, or religious construct).
1. Shift the focus from families to communities
Monogamy in the terrestrial animal kingdom is fairly common. Animals pair up and stay paired up at least for the season, but often for life.<citation needed> This isn't because animals have developed complex philosophies and religions. It's because a bonded pair can more easily defend themselves against the vagaries of life. It's more common to defend one's family from various problems within your own community than it is against those from another community.
But let's reverse that on your world! Let's assume, for the sake of argument, that shared breeding produces a community-specific marker. let's say a common pheromone. You can literally identify which community a single member is from because that member has a smell that's shared with every member of the community. And each community's smell is different. In this case, competition is believably shifted to between communities. Consequently, it's favorable for females to bear as many children from as many males as possible to promulgate and strengthen the community. The tendency would be to promote greater harmony within the community because competition has been shifted to between communities.
That genetic marker can be anything. A smell, a color, a size, an extra toe... anything, so long as it creates the us vs. them condition that evolution requires to, well... promote evolution.
2. Let's change the nature of conception to shift the idea of children being "mine" to being "ours"
Here on Earth one issue driving evolution is that children are a limited resource. For example, human females are good to have 2–6 children, but rarely have 6–12 and very rarely more than 12. Children are a fairly rare commodity. The chance to propagate who you are has a narrow window for success. From an evolutionary perspective, the drive to be a part of that propagation is very strong, leading to competitive selfishness for mates.
But what if your females can have more children, remain fertile longer, or both? Suddenly the pressure to be the only propagator is less. This has consequences, like really large populations, which could be moderated by shortening the average lifespan... but maybe there's a better idea.
What if we reverse the nature of conception? Here on Earth females have a narrow window for bearing children (fixed number of eggs, eggs available only periodically, etc.) but the males can fertilize eggs basically starting from puberty and at-will until they die. So, let's reverse it. It doesn't matter if you change how many eggs your females have or how often they're available for fertilization—just change the male's contribution to something that's infrequent or even rare. Using Earth's humans as an example, the idea is to change the male so that he isn't available/able to fertilize eggs every month.
Suddenly the species requires polyandrism just to survive.
3. On your world, immunity to disease found a different path than here on Earth.
Finally, here on Earth evolution solved the problem of disease by creating an absolutely amazing immunity system. While some immunity is passed from mother to child, most of the system is designed around the individual's ability to internally detect, analyze, and defeat disease.
But what if on your world it took another path? Rather than the direct attack method we Terrans use, what if your world evolved an innoculation-based solution? In other words, the immunity passed mother-to-child is much more important in your world than here. The idea is that the chance of any individual developing a resistance to disease is much lower than here on Earth—but the ability to transmit that resistance to the next generation is much higher than here on Earth.
Which means the protection of the species depends on females mating and conceiving with as many males as they can. A negative is that epidemics will be more immediately devastating because it takes at least a generation to push the solution through the ranks. But that would lead to evolution favoring females that conceive easily, bear quickly, and have the higher tendency to healthy offspring.
(In this case, limitations on the number of bonded males would come about through the development of civilization, law, and the tendency for one person to think they have the right to stick their nose into another person's business).