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I don't normally do bidirectional tracks, but I believe the problem is that your signals are too close to the fork. Trains do not stop on a dime, they need some space to stop, so if one train approaches the block quickly from the left, and another is entering slowly from the top base first, the one moving quickly from the left cannot stop in time to avoid collision.
To avoid this, place the signal for the base further back into the base so it has priority getting onto the track. This way it reserves the block earlier, before the other train can just barrel through the intersection without being able to stop. It may be suboptimal for throughput, but it should stop collisions.
Another option is to put in a switch track at that intersection for trains moving west to east that cannot stop.
If you have only one long block, then only one train can be on that section of track at a time. If you break the long track runs into smaller segments, you could have two or more trains traveling on the same track at the same time, one behind the other. That would also give an addtional place for a train to stop before it gets into a problem area.
I recommend that you eventually go to single headed trains, with a double track mainline and branches off of that to your various areas. It requires roughly twice as much track, a bit more planning and a bit more room. But, it is infinitely expandable with minimal work. And you can have trains heading out "into the field" and back to the base at the same time.
This is why you don't put signals right at the merges.
I have never used double headed trains, but they work the same as single headed trains with regards to signals. I am a train nut in Factorio, with many hundreds of trains in my bases. Not once have I EVER had two trains collide unless I was manually driving one.
I know trains don't stop instantly (in fact, there is infinite research to increase the braking power), but as long as there are some signals I have always seen trains stop before an impact can occur. Whether they just sit at the station, or stop before the problem area. When a signal is red, a train will start braking well ahead of that signal because it knows its own stopping distance. Additionally, if a signal is red, another train will not attempt to enter that block.
Perhaps two trains are entering one block at precisely the same time? That'd be an interesting experiment.
Train A is moving at 30 kph speeding up entering intersection from the north right now, it has now reserved the block and not before!
Train B is moving at 200 kph 3 meters from rail signal to the west of intersection right now, and that is 10 meters from the intersection.
Train B needs 50 meters to stop. Train A will be in the intersection in 2 seconds, but train B is going to be in the intersection in 2.5 seconds, as it cannot stop for the rail signal that just turned red.
The reservation for the intersection does not happen until a train actually enters the rail block. If the slow train enters first, and the fast train cannot stop, it simply cannot stop. To counter this, move the rail signal back for the SLOW train back far enough that the fast train is signaled early enough that it can stop.
You have too many signals in that spot in my opinion, I would reocmmend removing a lot of the signals that lead down to the bottom platform, like the image attached so it is one block instead of many. I even have chain signals shown in spots I think would benefit it, and would recommend you having your exit be similar into a 2-line branch for your areas where one line is going one way and the other is going the other way. https://steamcommunity.com/sharedfiles/filedetails/?id=2214031974
Problem could probably solved if all except those leading into a dead end station are substituted by chain signals (2 lights).
Even if there were no collisions, there should be a dead lock sooner than later...
After all two-way-rails are an intersection and the golden rule for signals still is:
Chain signals before & in intersections and normal signals after intersections & on straights with space for train length.
Nonsense - that is not how Factorio trains work at all: Automatic trains reserve all blocks they would need to stop, and no other trains are allowed to enter these blocks. As a result, it is only possible to get collisions if you either 1) do not use any signals at all or 2) manually drive a train.
If OP thinks that they are getting collisions in automatic mode then they are simply wrong. He will get deadlocks due to incorrect signalling but no collisions.
https://imgur.com/a/PVL6I/all
https://www.youtube.com/watch?v=OH8L_hTIIJg&feature=youtu.be
Oh, ABSOLUTELY!! Yes! Yes x1000!