This is a very rare problem, happening on average once in every 35,000 pregnancies. One twin is usually completely normal. The other is body-like tissue, often with legs and a lower body, but no upper body or heart. Abnormal blood vessels on the placental surface allow the normal twin (aka pump twin) to pump blood though the tissue of the abnormal one. Because the pump twin heart has to pump for two, there is a high risk of going into heart failure. This would then lead to death of the normal twin, unless it is delivered if it is far enough along in pregnancy.
The risk of the normal or pump twin going into heart failure and dying seems to depend on the size of the acardiac. The larger the acardiac compared to the pump twin the greater the risk. The amount of blood flow into the acardiac also seems to play a role. The more blood flow the higher the risk. The harder the pump twin's heart is working, the greater the risk of heart failure also. All of these things can be looked for with ultrasound tests. In critical cases these tests may have to be repeated frequently.
One option in this case is simply watching for the earliest signs of heart failure in the pump twin, with frequent ultrasounds. If heart failure is identified, and the pregnancy is far enough along then the pump twin would simply be delivered.
The other option, if the acardiac twin is large enough and we are worried about the amount of blood flow to it causing heart failure in the healthy twin is to stop the blood flow with Fetal Image-Guided Surgery (FIGS-IT). We currently do this using a thin needle, which we guide into the place where the blood vessels feed into the acardiac twin, using ultrasound. Once in place this instrument called an RFA (radio-frequency ablation) device produces a very high local heat, to burn the tissue and destroy the blood vessels to stop the blood flow. The needle is so thin that no incision is necessary and the pain and recovery are similar to an amniocentesis. We were the Center to pioneer this treatment and have had great success with it.