In addition to the previous answers, this thing is completely obvious in Lattice gauge theory. This shows that axial and temporal gauges are well defined, and also gives a full gauge fixing prescription removing residual gauge ambiguities. It is the lattice version of Qmechanic and Bar-Moshe's answer.
On a lattice, there is a group element for each lattice link. There is a freedom to multiply by a group element at any point. So to completely fix a gauge, all you have to do is choose a unique path to this point from some starting position, and this gives you a unique group element at each point to multiply by.
Make this position the origin, and define the path as follows:
follow the x-axis to the x-coordinate of the point, then follow parallel to the y-axis to the x,y coordinates of the point (still at z=0 t=0), then follow parallel to the z-axis to the proper z, then parallel to the t-axis. Multiplying all the group elements in the order you encounter them. Then rotate by this matrix at this point.
This sets the gauge group element in the t-direction to the identity, this is equivalent to setting the continuum gauge field to zero. It also sets the gauge field in the z-direction to zero on the t=0 surface, it sets the gauge field in the y and z direction to zero on the x-y plane, and the entire gauge field to zero along the x-axis.
This is a complete and nonperturbative lattice gauge fixing, it leaves only the global gauge group unbroken. It corresponds to axial gauge, because it is in imaginary time, but to do the temporal gauge fixing heuristically without regulator is identical as explained by Qmechanic, and there are no problems of principle (although, as Bar Moshe said, you have to be careful not to throw away the constraint equations of motion that arise from varing the gauge field the directions that violate the gauge condition).