Modern search applications feature real-time as-you-type query search. In its elementary form, the problem consists in retrieving a set of k search results, i.e., performing a search with a given prefix, and showing the top ranked results. In this paper we focus on as-you-type keyword search over social media, that is data published by users who are interconnected through a social network. We adopt a "network-aware" interpretation for information relevance, by which information produced by users who are closer to the user issuing a request is considered more relevant. This query model raises new challenges for effectiveness and efficiency in online search, even when the intent of the user is fully specified, as a complete query given as input in one keystroke. This is mainly because it requires a joint exploration of the social space and traditional IR indexes such as inverted lists. We describe a memory-efficient and incremental prefix-based retrieval algorithm, which also exhibits an anytime behavior, allowing to output the most likely answer within any chosen running-time limit. We evaluate our approach through extensive experiments for several applications and search scenarios. We consider searching for posts in micro-blogging (Twitter and Tumblr), for businesses (Yelp), as well as for movies (Amazon) based on reviews. We also conduct a series of experiments comparing our algorithm with baselines using state-of-the-art techniques and measuring the improvements brought by several key optimizations. They show that our solution is effective in answering real-time as-you-type searches over social media.