Tiger and Monkey compatibility
Tiger (Yin, 寅) and Monkey (Shen, 申): a clash (Chong, 冲) 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
Clash (Chong, 冲)
Tiger and Monkey sit directly opposite each other on the circle of twelve, six signs apart. The book calls clashes broadly ill, and then names the ones that are not.
Among the branches the seventh along is the clash, just as among the stems the seventh along is the killing force.
地支取七位为冲，犹天干取七位为煞之义。
A clash is the twelve branches at war, striking at each other, and broadly it is ill. Yet some clashes bring the greatest blessing: those where the two sides also feed each other.
相冲者，十二支战击之神，大概为凶。然有为福之甚者，乃冲处相生。
So a clash or a break can be good or bad. They cannot be judged wholesale.
可见冲破有吉有凶，不可概论。
San Ming Tong Hui, volume 2: On the clash (论冲击) · source text
Punishment (Xing, 刑): the punishment of no gratitude
Tiger 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
Their elements
The Tiger is Wood and the Monkey is Metal: Metal controls Wood.
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
- Tiger: 1926, 1938, 1950, 1962, 1974, 1986, 1998, 2010, 2022. In 2027, the Year of the Fire Goat (Ding Wei, 丁未), the Tiger 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.
Tiger with the other signs: Rat · Ox · Tiger · Rabbit · Dragon · Snake · Horse · Goat · Rooster · Dog · Pig.
Monkey with the other signs: Rat · Ox · Rabbit · Dragon · Snake · Horse · Goat · Monkey · Rooster · Dog · Pig.