Snake and Monkey compatibility
Snake (Si, 巳) and Monkey (Shen, 申): a six harmony (Liu He, 六合) and a punishment (Xing, 刑).
These are fixed relations among the twelve branches of the Chinese calendar, set out in the classic San Ming Tong Hui (三命通会), written in 1578, quoted below. They describe the signs, not two people: no classic gives a method for matching two people by their signs, so there is no score here.
What the book says
Six harmony (Liu He, 六合)
The six harmonies pair each yang sign with a yin one. Snake and Monkey are one of the six pairs. The book’s explanation is in numbers: each sign has a place in the count of yin and yang, and a pair adds up to three or nine.
A harmony is a joining: yin and yang in accord, so that their qi joins of itself.
夫合者，和也，乃阴阳相和，其气自合。
Rat, Tiger, Dragon, Horse, Monkey and Dog are the six yang; Ox, Rabbit, Snake, Goat, Rooster and Pig the six yin. So one yin and one yang in accord are called a harmony.
子寅辰午申戌六者为阳，丑卯已未酉亥六者为阴，是以一阴一阳和而谓之合。
The Snake is the sixth yang and the Monkey the third yin: six and three make nine.
巳为六阳，申为三阴，六三得九数
Rat and Ox, Horse and Goat each make three, and three gives birth to all things. The rest all make nine, the highest of the yang numbers.
子丑午未各得三者，三生万物，余皆得九者，乃阳数极也。
San Ming Tong Hui, volume 2: On the six harmonies of the branches (论支元六合) · source text
The book also quotes an older work, the Tian Yuan Bian Hua Shu (天元变化书), which gives the pair a line each way round:
Snake joining Monkey: the blessing is slow. Monkey joining Snake: the qi of office is abundant.
已合申福慢，申合巳官气盛
San Ming Tong Hui, volume 2: On the six harmonies of the branches (论支元六合) · source text
On what a harmony brings to a chart:
Wherever a six harmony or a three harmony enters a chart, it means a person of fine looks and settled spirit, who loves life and hates killing, straight of heart, easy to deal with, bright and quick to understand.
凡六合三合入命，主人形容姿美，神气安定，好生恶死，心地平直，周旋方便，聪慧疏通。
Broadly: a harmony with good stars is good, and a harmony with bad ones is bad.
大率合吉神则吉，合凶神则凶。
San Ming Tong Hui, volume 2: On the three harmonies of the branches (论支元三合) · source text
Punishment (Xing, 刑): the punishment of no gratitude
Snake and Monkey are two of the three signs, Tiger, Snake and Monkey, in the punishment of no gratitude: the book’s reason is that each turns on what gave it life.
Heedless of what gave them life, they hold each other down from a distance. So it is called the punishment of no gratitude.
不恤所生，遥相克制，故曰无恩。
Seeing a punishment, do not at once call it ill. Look whether the five elements hold a good hour, strength, the officer star, the seal, a noble spirit, virtue or blessing; with these good things to help, the punishment does no harm and is put to use instead.
凡见刑不可便以凶论，须看五行中有无吉辰、旺相、官星、印绶、责神、德福等物，有此诸吉相扶相助，刑不为害，而反为用
San Ming Tong Hui, volume 2: On the three punishments (论三刑) · source text
A break, in folk lists only
Folk lists also count Snake and Monkey as a break (Po, 破). San Ming Tong Hui says not to: these two are in harmony, so the break is not counted.
In this force, Rabbit and Horse, Ox and Dragon, Rat and Rooster, Goat and Dog all break each other. Only Tiger, Monkey, Snake and Pig break by origin yet are in harmony, so they are not counted.
此煞，卯与午，丑与辰，子与酉，未与戌，皆相破。惟寅申巳亥原破，却三合，故不取
San Ming Tong Hui, volume 3: On the symbolic stars as a whole (总论诸神煞) · source text
Their elements
The Snake is Fire and the Monkey is Metal: Fire controls Metal.
San Ming Tong Hui’s section on the zodiac animals happens to name these two. It gives the element reasoning, and then asks why the real animals do not behave that way:
The Snake is the snake and the Monkey the monkey. If Fire overcomes Metal, why does the snake not eat the monkey?
巳，蛇也，申，猴也，火胜金，蛇何不食猕猴？
Matched by how their elements overcome each other, the creatures of the twelve branches do not correspond at all. Wherever blooded creatures master one another, even to eating one another, it comes down to sharp or blunt teeth and strong or weak sinews.
十二辰之禽以气性相克则尤不相应。大凡含血之虫相服，至于相啖食者，以齿牙钝利、筋力优劣自相胜服也。
San Ming Tong Hui, volume 2: On the animals of the branches (论地支属相) · source text
How much it counts
A sign is one character of the eight in a BaZi chart, the year’s branch. The book reads these relations between the pillars of one chart, and tells the reader to look at the whole chart’s elements first:
Clash and break, three harmony and six harmony: where a chart has them, ask next how its five elements get on. Misfortune may give rise to blessing, and blessing to misfortune.
相冲相破，三合六合，命中有之，即求五行相得何如。或祸中生福，福中生祸
San Ming Tong Hui, volume 3: On the symbolic stars as a whole (总论诸神煞) · source text
Birth years, and 2027
- Snake: 1929, 1941, 1953, 1965, 1977, 1989, 2001, 2013, 2025. In 2027, the Year of the Fire Goat (Ding Wei, 丁未), the Snake has no clash with the year.
- Monkey: 1932, 1944, 1956, 1968, 1980, 1992, 2004, 2016. In 2027, the Year of the Fire Goat (Ding Wei, 丁未), the Monkey has no clash with the year.
A sign’s year is usually counted from Chinese New Year, between late January and mid-February; BaZi counts it from the Start of Spring, about 4 February. Born in January or early February? You may be the sign before: the sign pages check your date.
Two real people?
Each person’s chart has eight characters, and the same relations run between all of them, not only the two signs. The two-chart page sets both charts side by side, branch by branch, and gives what each chart says about marriage on its own. No score there either.
Snake with the other signs: Rat · Ox · Tiger · Rabbit · Dragon · Snake · Horse · Goat · Rooster · Dog · Pig.
Monkey with the other signs: Rat · Ox · Tiger · Rabbit · Dragon · Horse · Goat · Monkey · Rooster · Dog · Pig.