Rat and Monkey compatibility
Rat (Zi, 子) and Monkey (Shen, 申): two of a three harmony (San He, 三合).
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
Three harmony (San He, 三合)
The three harmonies are four groups of three signs, each four apart, and each group makes one element. Rat and Monkey are two of the Water group: Monkey, Rat, Dragon. The third, the Dragon, completes it, and the book says a group needs all three.
Take Monkey, Rat and Dragon. The Monkey is the Rat’s mother and the Dragon the Rat’s child: at the Monkey Water is born, at the Rat it is at its height, and the Dragon holds its qi. Birth produces, height completes, the store gathers in; with birth, completion and gathering, all things have a beginning and an end. That is the way of nature, and so Monkey, Rat and Dragon make the Water frame.
如申子辰，申乃子之母，辰乃子之子，申乃水生，子乃水旺。辰乃水气，生即产，旺即成，库即收，有生有成有收，万物得始得终，乃自然之理，故申子辰为水局。
If one of the three is missing, the change does not make a frame, and it cannot be read as a three harmony.
若三字缺一则化不成局，不可以三合化局论。
San Ming Tong Hui, volume 2: On the three harmonies of the branches (论支元三合) · source text
Their elements
The Rat is Water and the Monkey is Metal: Metal feeds Water.
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 monkey is afraid of the rat, and what bites the monkey is the dog. The rat is Water and the monkey Metal; Water does not overcome Metal, so why does the monkey fear the rat? The Dog is Earth and the Monkey the monkey; Earth does not overcome Metal, so why does the monkey fear the dog?
猕猴，畏鼠者也，啮猕猴者，犬也。鼠，水，猕猴，金也，水不胜金，猕猴何故畏鼠？戌，土也，申，猴也，土不胜金，猴何故畏犬？
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
- Rat: 1924, 1936, 1948, 1960, 1972, 1984, 1996, 2008, 2020. In 2027, the Year of the Fire Goat (Ding Wei, 丁未), the Rat harms the year (Hai Tai Sui, 害太岁).
- 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.
Rat with the other signs: Rat · Ox · Tiger · Rabbit · Dragon · Snake · Horse · Goat · Rooster · Dog · Pig.
Monkey with the other signs: Ox · Tiger · Rabbit · Dragon · Snake · Horse · Goat · Monkey · Rooster · Dog · Pig.