I AM A TOYOTA MASTER TECH, 15yr;
Synchronizer will not cause pop out, synchro is a braking device to match gear speed to shaft speed so locking collar can slide onto the mating teeth on the gear (no gear clash/smooth shift). The shift linkage on the "W 5X" family of trans is internal.
First:::is the shifter positive in finding the appropriate gate, (1-2 and 3-4 and 5-R),no excessive slop? First thing to do is lift the shifter out of it's pivot-ball socket. You will need to remove the shifter boot. Then you should see a cap stamped from steel that the shifter pass's thru into trans. Push down and turn that cap, remove shifter. Look at the bore, there should be a nylon socket, usually green, some are blue. Is it fallen apart or cup shaped? These wear, especially when older. Have to replace the housing, can't buy the bushing. I take bushing out of the new, install it to old housing. Don't have to deal with getting at housing in the truck to change it.
Other will be internal M/T problem, worn collar-hub-keys, excess gear lash on the shaft,or worn brg or bushing. Go with the shifter inspect first. Mail me your finding.