A while back, I accidentally dropped one of my favorite pipes on our hardwood floor. This is not a thing that I normally do, and I was annoyed with myself. Further, the stem snapped neatly off, with the tenon breaking in the middle. It left me with half a tenon stuck down at the bottom of the mortise.
Here's a little extraction trick, possible when your pipe's shank is thick enough to allow it:
Thread the wood screw in carefully, then grip it with pliers and tug the broken tenon section out. Voila!
PS - If you do this with a too-large screw or in a thin-shanked pipe and crack it, don't come crying to me. This blog fully supports the concept of Your Mileage May Vary, and Use at Your Own Discretion.
The next part was the experiment. I wanted to try some special plastics glue, a 2-part mix designed exclusively to bond "hard to glue" plastics with bonds requiring exceptional strength. I followed their instructions for preparing the broken surfaces, then applied glue and pressed. I carefully scrapped off the overflow, then sanded the area of the tenon break before re-polishing it. The split line is visible, but hard to see. Better yet, the bond is amazing - Thus far it has held up to quite a bit of experimental twisting and insertion.
Only time will tell whether the fix will last long-term, but it's certainly nice to have my pipe back again!