Tail latency from scatter-gather fan-out

Your search-results page (Java) issues a scatter-gather fan-out to 50 shard servers and waits for all 50 before responding. Each shard's individual p99 is a healthy 15ms, but the page's end-to-end p99 is 220ms and the p99.9 is over 1s, far worse than any single shard. There was no deploy. A node-replacement event this morning left a handful of shard replicas serving from cold caches and one replica on a host with a noisy neighbor. Explain the math, triage, and propose mitigations.