Right, when run in a fully asynchronous setting, where the network adversary can always deliver messages in the order of their choice, the FLP impossibility result says that Paxos cannot terminate.
Thus, for Paxos to make progress, the network needs to eventually "heal" and deliver messages in a timely manner -- that's what we mean by partial synchrony. Streamlet, likewise, tolerates arbitrary network delays (in the sense that no two players will agree on different things even if Comcast maliciously scheduled every message), but also requires partial synchrony to make progress (i.e. once in a while Comcast will deliver messages in timely fashion and in order, like you expect).
(Partial synchrony is arguably the most practical of the theoretical synchrony models for consensus. There are other models, i.e. "periods of synchrony", but they all end up capturing the same idea).
FLP, however, is a fairly weak statement. It only rules out termination for deterministic protocols. We can in fact design randomized asynchronous protocols that make progress (with good probability), regardless of the ordering in which messages are delivered. I.e. as long as Comcast eventually delivers every message -- and letting it choose delays arbitrarily -- a randomized protocol could make progress. These are the "true" asynchronous protocols. (Paxos is not one of them)
I guess the original asynchronous consensus protocol is Ben-Or's protocol (which takes 2^n time in expectation IIRC, where n is the number of players). We can reduce the running time using some (fairly sophisticated) cryptography -- the most famous is the Canetti-Rabin protocol, and Cachin and many others have worked on it since. Most of the complexity (for polynomial time protocols) is in building a "global common coin" -- i.e. a global coin that everyone agrees on the result of; once you have that we can use Mostefaoui's protocol for instance. Building this coin is incredibly complicated (much more so than Paxos) unless we assume some strong cryptographic trusted setup (i.e. a nice threshold signature scheme). How practical any of these schemes are is an open question.