Virtually every stock market is accompanied by a parallel "upstairs" market, where brokers take calls from large traders and help them to locate counterparties and negotiate the terms of a trade. Even after the switch from open outcry to mostly computerized markets, upstairs markets still exist. They are based on relationships. Upstairs brokers seek out counterparties mostly from traders who are known to them. They then negotiate transactions that might otherwise be executed at an inordinate cost or delay.
Electronic trading lowers the cost of trading for relatively liquid stocks in block sizes unlikely to overwhelm the market. However, parties who wish to make large block trades, or trade in illiquid stocks, frequently use the upstairs market. 
- The "upstairs market". allofinancier.com.