I'll add to @wombie answer that doing so would be quite dangerous.
When an http agent (say a reverse proxy or a load balancer, acting between the browser and the final server) receive a response it usually assume that the query was fully sent and that the next query can be send back.
Most reverse proxy agents would not use pipelines when transferring queries to a final backend. That means even if the browser is trying to do a pipeline of n queries the communication between the reverse proxy and the backend will usually be n individual queries and responses, waiting for each response before sending the next query (could use a keep alive connection, but usually not pipelining).
This behavior greatly reduce the risk of HTTP smuggling, with HTTP requests or responses attacks.
Sending your response before receiving the whole body will facilitate the smuggling attack.
To be more clear, imagine you are sending a single query, with a strange syntax and a big body. This body contents looks like another query with a body. And the body of this query-in-the-body is another third query. But this is just one query with a body.
The reverse proxy has a bug on the strange syntax and does not see the body of the first query. For the proxy it's a pipeline of 2 queries. So it sends this first query, without the body, to the backend.
The backend server has no bug on the strange syntax, so it's waiting for the body. But it sends an early response... we'll start having problems. If you'd wait for the complete body before sending the response the attacks would have stop here.
The reverse has a response, so it thinks it can proceed to the next query, and sends back the pseudo-second-query (which contains a body with an hidden third query) to the backend.
The backend thinks it's just the body of the first query, and does something with it, or not. But for the backend it's not a query. Let's say this body has a size of 20k, and that the backend is waiting for a body of 10k. The last 10k is another query for the backend, a query hidden in the body, and this query will be treated as another one, with a new response.
The proxy will receive a response for the hidden third query, and believe it's a response for the second query.
You may exploit this for cache poisoning. The early response is not enough to build an attack (here you also need a strange syntax on the query where the proxy and the backend will have a different perception of the body size), but it makes exploitations more easy. And, of course, you can find this behavior in the wild.