Hi - Xapian developer here, and Mongo DB enthusiast (though I've not had an excuse to play with it in anger yet). I'd like to help make a tight integration between Xapian and MongoDB, if there's interest in it. I'm not quite sure what the best approach for linking would be, though.
Xapian certainly supports "realtime" updates in the sense described above. It also has some features in trunk for supporting replication of the index, which might be helpful when working with MongoDB.
One basic approach would be to hook into the updates in Mongo somehow, and send them across to a parallel Xapian index for full-text indexing. I think this would be best done by defining some kind of schema, though: often, when searching, you want to search across a fairly complex set of fields (a common example is to search across both title fields and content fields, but to boost the importance of the title fields - but in real world search situations, you often come up with much more complex requirements). A naive mapping of a particular field in Mongo to a search index would allow basic search, but we can do much better than that, I think.
There are also things like thesaurus entries and spelling correction, which you would want to be able to configure somehow.
Mongo doesn't really have schemas yet, IIRC, so I'm not sure how the Mongo developers would feel about adding that sort of context.
When defining searches, Xapian has a built-in and flexible query parser (which is aimed at parsing queries entered into a search box by the average untrained user, so supports some structure (eg, field:value), but copes with any random input in a sane way). It can also have structured searches built up, and combined with the output of parsing several user inputs, so a mapping from a Mongo-style query to a Xapian search could be defined to limit Mongo results.
Xapian also has things called "External Posting Sources" which are arbitrary C++ classes (subclassing a Xapian::PostingSource class), which can be used to perform combined searches across data stored in the Xapian index, and external data. (A "posting source" in search engine terminology is a list of documents matching a particular word (or term) and is the fundamental piece of data stored in a search engine index.) This could be used to limit searches to documents matching a MongoDB query pretty efficiently, without having to store extra data in Xapian.