The short answer is yes, you can, but not with catches.
First about watching in realtime: HDHomeRun cannot directly stream to PS3, unless there are customized SW on PS3 to support it. Realtime transcoding is theoretically possible, but I guess you'll have to make sacrifice in quality.
Now about recording it, post process it, then watch; there are many ways to achieve this: ATSC broadcast is basically MPEG-2 video stream and AC3 audio stream in transport stream encapsulation; now both MPEG-2 and AC3 are supported by PS3 already, but I'm not sure if PS3 is capable of reading TS record streams directly. But with post processing, you can change the encapsulation to program stream, or even re-encode it in AVC.