Dual power supply without center tapped transformer
You can create a bi-polar supply from a transformer without a center tap as long as you can live with just half wave rectification for the positive and negative supply rails. 

Here is a schematic that shows it is feasible. Note that I left out the filter caps so that you can see the half wave rectified waveforms. 



Here are the waveforms of the two output voltages:



Half wave rectification places some requirements that you add bigger capacitors to the output to smooth the DC voltage to less ripple than would be needed if you did full wave rectification. It can also be beneficial in this case to use a higher transformer secondary voltage so that more ripple on the capacitors would not cause your down wind voltage regulator to drop out cycle by cycle. 

Note that it is conventional to do full wave bridge rectification on an untapped transformer secondary. Unfortunately you cannot get two bridge rectifiers to work on a single untapped transformer secondary to get a bipolar output.