How is this precipitate possible?

When you provide the correct information - you get an answer:

Go through each step and determine what happened:

1) We started off with (both .1 M and 10 drops of each)
Na2CO3 + AgNO3
What took place:
Na2CO3(aq) +2 AgNO3 (aq) → Ag2CO3(s) + 2NaNO3(aq)
You should have seen a precipitate of silver carbonate. You have seen this, in your comments

You have NaNO3 in solution and Ag2CO3 as a precipitate

2) Afterwards we added the
6 M HNO3 (about 20 drops or so)

What took place:
When you added the HNO3, you formed a very acidic solution: Carry down the products of the first test and add HNO3 - what do you get:
Ag2CO3(s) + 2NaNO3(aq) + HNO3(aq) → 2AgNO3(aq) + CO2(g) + H2O(aq) + NaNO3
The precipitate of Ag2CO3 reacts with the HNO3 and dissolves to reform AgNO3. The CO2 gas escapes to atmosphere and the H2O just forms part of the solution.
You have seen this: after adding the HNO3
the precipitate cleared (it formed H2O (l) + CO2 (g) )

Now you are left with a solution containing only: AgNO3 and NaNO3

3) then we added the
.1 M HCl (about 17 drops)

What took place:
If you add HCl to a solution containing AgNO3, you get a precipitate of AgCl
AgNO3(aq) + NaNO3(aq) + HCl(aq) → AgCl(s) + HNO3(aq) + NaNO3(aq)
You have formed a precipitate from the reaction of the AgNO3 with the HCl. The NaNO3 is carried forward from the start of the test and does not really play any role, but it is there so I show it.
You saw this:
then after adding HCl
it formed like a white layer of precipitate

To answer your question: The final precipitate resulted from the reaction between the HCl and the AgNO3 in the well.

The CO2 and the H2O formed have nothing to do with the formation of the precipitate.

If you formed a precipitate upon addition of HCl, then there is something missing from what you've given us. Are you sure there wasn't some Na, Mg, Ca or something else in there?