This is an interesting puzzle yet a bit incomplete as it only covers two players out of the 4.
Here are the 4 players who play a key part in the daily activity of markets. (FIIs, DIIs, Pros and Clients)
Source: Sensibull
Note: There are a few more key players like Govt. & Promoters etc. but we can ignore them for now)
Our General Misconception
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Generally, Most of the traders are wired to study and mainly focus on FII & DII cash data and arrive at a conclusion about whether markets are bullish are bearish. Even in this, We only have data from 2 players out of 4.
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More importantly, This is related to the cash market whose outcome of profits or losses is medium-term in nature generally. While there is some indication of the trend, it is almost negligible.
What should a trader keep an eye on
- For a trader, It makes sense to focus more on F&O data to get more clarity on short-term trend.
We have started sharing @Sensibull’s overnight analysis of F&O and overall market analysis on TradingQ&A. In my opinion, Abid’s incredible insights are super handy for a trader 
The Original Question - What’s the secret between the FII & DII coordination in Cash Market ?
While knowing the exact mechanism as to how this works is not so easy, In my opinion, two key factors that i can think of right now, might determine this :
1. The general strategy of FIIs and DIIs. Over the years, We have seen FIIs are momentum players who buy high and sell low and DIIs follow the contrarian style where they sell when the market is higher and buy when there is a fall.
A key point to be noted here is that as there are many players in both FIIs and DIIs, it doesn’t make sense to make simplistic conclusions like all FIIs are bullish and all DIIs are bearish. If there is an FII whose bullish view matches the bearish view of another DII, the coordination that we are talking about might happen.
2. The role played by other players like Clients & Pros. The numbers contributed by these players in both cash and F&O markets help in balancing the figures without affecting the day’s trend as you rightly asked.
To give you an example of this, We will see that many a times, Markets will see a 2% fall, but FII and DII net sell and buy figures are negligible. so what might have caused the fall? The activity of FIIs, Pros, and Clients in the F&O segment.
That being said, this is a nice topic to dig further deeper into, We will research a bit more on this and share if we find anything interesting.