Leave USA before getting actual green card

Angelica writes, "My son sponsored me for a green card and I have been told by the USCIS that no interview is needed for my husband and I.  The letter added that our alien registration cards will be mailed to us after our applications are processed.  Now, due to an emergency situation in my native country, I must leave the United States.  Since my son is also scheduled to visit his native country, when he travels there he can bring the actual green cards with him and then we can all come back.  Since we were not really expecting to travel out of the US, we did not apply for the advanced parole (AP) and now we do not have enough time to wait for it.  Do you think there will be any problem if we leave the US and then try to enter as green card holder?"

Actually, yes.  Most likely scenario is that you will not be allowed to enter the United States because your green card will not be valid.  By leaving the country (the USCIS will know from the information provided by the airline) prior to your green card being approved (you will either need a I-551 stamp in your passport or actual green card in your possession to really know for sure that you have been approved because Form I797C is not always enough), it will be concluded that you have abandoned your green card application and your adjustment of status was null and void.  When you present your green card to the agent at the airport, it will be repossessed and you will be sent back on the next flight.  The only loophole that exists is if you cross into Mexico by road and then take a flight from there (USA has no exit checks and at the land crossing into Mexico, there is no immigration check of entries from the American side), there will be no evidence that you actually left the United States prior to getting the actual green cards (while the probability is remote that someone will actually check that seriously, but if your passport has stamps of travel to and from other countries that show that you actually were not in the United States on the date of your approval, you will suffer the same fate).  In conclusion, if you value your GC, try to deal with the emergency remotely.