There are some good answers in the thread already, but thought it might be nice to collate them into a more complete and structured response.
Ultimately, there have been a range of solutions to this problem over the years for various scenarios and platforms. I've grouped my response by platform type just for the sake of readability.
Mobile platforms (but not towed or attached your ship)
Sonobuoys can transmit sounds over radio in real-time and have a long history for whale research (see e.g. McDonald et al 2003). They are usually temporary devices with a lifespan measured in hours, and if used this way, then lots of them are required to be deployed in succession for sustained monitoring over longer time periods. However several studies over the years describe how they can be repurposed into a reusable form (e.g. Horsley 1989, Van Parijs et al 1998, Rankin et al 2019). Alternatively, there have been some studies that have achieved the same effect as a reusable sonobuoy, but have started from a clean slate (e.g. Baumgartner et al 2008 comes to mind).
More recently the idea has been extended to autonomous vehicles, with on-board near real-time processing and satellite communications. Usually to reduce the amount of data sent over the expensive satellite link only a summary or portion of the detections are sent back in near real-time (e.g. Baumgartner et al 2021, Johnson et al 2022, Kowarski et al 2020).
Fixed platforms
Fixed platforms include moorings and cabled arrays.
Moorings are conceptually the same as the mobile platforms, but with an anchor (and all of the benefits and limitations that brings). They can have a terrestrial or satellite radio link, and can transmit 'raw' acoustic data or just detections. There are some commercially available products in this category from SMRU Ltd and Jasco that have already been mentioned in this thread. There are also custom scientific solutions (possibly commercially available, but a bit unclear) like the WHOI/Cornell BRP NARW moorings mentioned in Van Parijs et al (2009) and Baumgartner et al (2019).
Cabled arrays/observatories are pretty rare due to high costs of running submarine cables and servicing sensors. The main example that comes to mind for me for bioacoustics is the CTBTO, who operates several cabled observatories around the world to monitor for nuclear explosions. While I believe they can monitor in near-real time, I think most bioacousticians end up accessing their archived data rather than any real-time data stream. Additionally, the International Quiet Ocean Experiment has a public website with a list of some other cabled observatories that's worth checking out.
Though cabled observatories can be quite expensive, there has been some work in recent years towards repurposing existing undersea fibre-optic cables for underwater acoustic monitoring. Buoffaut et al (2022) provide an example, of how this can be done in real-time.
The above is not at all a fully comprehensive list, but should hopefully serve as a useful summary.
