How does this LED string work?
I expect there is an ordinary diode across the two pins of each LED such that when driven in the forward direction the LED lights and in the reverse direction the diode conducts. This diode is hidden in the black tape under the LED.

If a series string of these LEDs is formed where the one color (eg blue) is connected one way and another color (eg yellow) is connected in the reverse direction then the blue will all illuminate with positive current and yellow with negative current.

Two of these strings can then be assembled with a common return wire resulting in 3 wires in total. (put different colors in the second string).

The controller has to provide one polarity or the other depending upon the combination required.

The AC supply provides a convenient way of providing the two polarities.

Since each LED will be driven with 60Hz, flicker is normally fairly noticeable in the LEDs, especially if you move your head (or the LEDs) when viewing them.