In the image below, my problem is shown; I want to tangle two shapes. I had hoped that arranging individual anchor points would be the solution, but that is not the case. Is is possible to make it 'tangly' without creating extra shapes for the purpose of overlapping?
Unfortunately, no. A single shape can only have a single z-value / stacking order associated with it.
You will have to clip the shapes in different pieces and arrange those pieces to create the illusion of tangling. A very useful tool to use with that is to copy a shape and use Ctrl / cmd + F to paste the copy exactly in front of the original. You can also use Ctrl / Cmd + B to paste in back.