Is it possible to use the triangulation method to determine a fish position by ultrasonic telemetry, using a network of fixed receivers, without synchronizing them?
I believe it would be technically possible but it would be fraught with error, the whole purpose of the syncronisation is to account for clock drift between receivers. Without synchronisation you would be prone to timing errors causing you to inaccurately position your fish. Undoubtedly someone else here knows more about VPS and other similar systems than me, but I would stick to the commercial solutions for now.
it wouldn't be worth doing it without synchronising them, and even if you do so you could only get triangulations in very low complexity environments. Any rocks/boulders/coral cause acoustic shadows and echoes which will completely destroy the reliability of the triangulation. See welsh et al. (2012) http://link.springer.com/article/10.1007/s00338-012-0892-1
I agree entirely with Niall and Christopher, especially with regard to environmental complexity rendering triangulations useless. I tried what you are suggesting a few years ago in a controlled experimental situation, and the triangulations were nothing short of ridiculous even in a low complexity habitat (sand flats). I think the latter attests to the importance of synchronising receivers prior to deployment.
In theory this seems like a great idea. However, in actual practice the environmental factors that will cause acoustic shadows, echoes, etc. will would make your data near impossible to dechiper.
If the receivers themselves are not synchronized, you can also use "sync tags" to help correct for error (and maybe this is what you meant by synchronization to begin with, and if you did, my apologies).
Even if the receiver clocks differ, if you strategically place tags through your array (preferably ones with a fixed, known interval), you can later correct the clocks by seeing when the detections hit each receiver. Note that your receivers will still need to be fairly close together in time (shorter than the time between successive signals from your sync tags).
You need to ensure you have enough sync tags to ensure all receivers, and all possible triangles, are within detection radius of at least one sync tag.
Still definitely best to sync the receivers AND have synchronization tags (sync tags are especially important in long-term studies, as this provides more time for the receiver clocks to drift).
With VPS, we use sync tags with *pseudorandom* transmission sequences to remove the requirement for the receivers to be well synchronized in the first place.
Pseudorandom transmit sequences make it much easier to unambiguously correlate detections at pairs of receivers than fixed-delay transmissions when there is significant inconsistency between receiver clocks.
For more information about how sync tags are used to synchronize receiver time during post-processing in VPS, check out "Understanding HPE in the VEMCO Positioning System (VPS)" by following the link. Go to page 14.