Electronic – Brushless DC motor runs fine in one direction but not another

can FOC even work for BLDC without an encoder but just with Hall sensors?

You have an encoder, it just doesn't have very high resolution. But how to increase it?

Once you know how fast the rotor is spinning (time between Hall sensor signal changes) you can predict intermediate angles. If motor speed is changing then the prediction will be off, but you can compensate for this too by measuring the acceleration and factoring it in.

You could implement back-emf as a backup control system to aid detecting fault conditions matching sensor data and rotor position calculated by back-emf, thus being able to detect even those fault conditions producing a "good looking" set of values, and eventually bring the motor to a soft stop using BEMF for rotor position detection.

Another option, if you want to go the fault detection way, is to implement redundancy in your sensors and communication line. This can be implemented in various ways and probably is the best way to add a functional fault detection that not only detect faults but also enables you to not have to immediately halt the motor.

If both such solutions seems too complicate you probably are in a case were a serious fault detection algorithm is not needed and stopping the motor in case of incoherent data is "just the right solution" so you can fix that cold soldered hall sensor before finals :-)

As far as fault detection goes is probably more important to focus on current detection.

Have fun! :-)