But what you can do is create a proxy for adultery, and tightly control that, instead.
The ideal proxy is contact between men and women, because no contact = no adultery. After all, contact between men and women is much easier to police than physical intimacy.
Interestingly, to limit contact between men and women, you only need to limit the freedoms of one gender. And, by "one gender", I mean women. Historically, in the real world, it has invariably proven much easier to limit the freedoms from women than of men. I'm told that it has something to do with reproductive economics, with power, and with tradition... but you can have whatever reasons you like in the world that you're building.
To limit the freedom of a gender you typically need rules. In fact, you need several kinds of rules.
First, you need Enforcement Rules. These are intended to convince people that:
- These rules are for the protection of women
- These rules are sacred
- These rules are tradition
- These rules are for the benefit of all society
Second, you need Punishment Rules. Something like:
- If any man or woman breaks these rules, they will be punished. They will be tortured and put to death by the community, in a public place. By flogging, stoning... something brutal, anyway.
And then you need Separation Rules. They need to say something like:
- Females over 10 years old must not go out alone
- When females venture out they must be accompanied by
a brother, father, husband, or another woman
- A female accompanied by one other person is to be looked on with suspicion. Such a woman is shameful, although leeway is given if her circumstances allow. A female accompanied by many other women is a credit to her community. She is to be highly esteemed.
- It is the duty of all females to accompany either women when the go out.
- If a woman lives alone, or there is nobody in her household who can act as her chaperone, she may travel alone to her nearest female neighbour
- If a female is seen with a male who is neither her father, brother, nor husband, then both male and female will be put to death
- When females go out, they will be covered up. Their face must not be visible, neither their arms, nor their legs, nor any other part.
Finally, you'll also need Marriage Rules. A society still needs men and women to meet and marry, so you need something like:
- Elderly women will act as matchmakers and chaperones, so that men and women can meet and marry
- No divorce. Ever. On pain of death.
You also want to include a couple of Random Rules (so you know that an ineffable god is involved, somehow).
Of course, you can still put adulterers to death if they're caught by eye-witnesses.
(Again, it appears that it is is often easier to catch women in the act of adultery, than it is to catch men; a well-know example being the woman caught in adultery in John's Gospel. Perhaps this is because people are more likely to turn a blind eye to male adultery. Perhaps it is because, when "evidence" turns up 9 months later, it points very clearly to a specific woman, but not so clearly to any particular man).
My intent in answering as I have appears to have been misunderstood by at least one commentator. Let me be crystal clear: I am absolutely not promoting the "keep women under lock and key" approach as a solution to any real-world problem. In the real world, I find such an approach utterly abhorrent.
Just as the OP is interested in a fictional world where adulterers are punished, I am proposing a fictional approach to bringing about a (more-or-less) similar result.
Handled well, the fictional oppression of any group in society can drive moving narrative, and can underpin compelling high-stakes drama. At the same time, fictional oppression can be a great vehicle for thought-provoking social commentary. It can encourage debate, and (I hope) it can help create a cultural environment where positive social change is possible.
I'm thinking of Malorie Blackman's Noughts and Crosses, for example.
More pertinently, I deeply admire the way that Margaret Atwood set The Handmaid's Tale in an extremely misogynistic society. She has my admiration, not because I endorse the society that she wrote about, but because it drew my attention to the appalling mistreatment that some women face in the real world (and because it enabled her to tell a rattlingly good story at the same time).
I hope and pray that I am a better human being for having read Atwood, Blackman, etc. and that fictional oppression can be a vehicle for reducing oppression in reality.