Asylum: Entering the USA Without Refugee Travel Documents

The USCIS lays things out here:

If you are an asylee and returned to your nation of origin:

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Quoting What Happens if a Asylee Returns to the Nation From Which She Seeks Protection

If you are an asylee who applied for asylum on or after April 1, 1997, then your asylum status may be terminated if you return to the country from which you were seeking protection See the “Fact Sheet: Traveling Outside the United States as an Asylum Applicant, an Asylee, or Lawful Permanent Resident Who Obtained Such Status Based on Asylum Status” to the right for more information.

Whether a refugee or asylee, if you accrued unlawful presence:

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Quoting Refugee Travel After Unlawful U.S. Presence

If you have been in the United States illegally, then you may be subject to a bar to admission if you depart the United States, even if you have been issued a travel document. For more information please see Section 212(a)(9) of the Immigration and Nationality Act (INA).

Beyond that, it's going to turn on the discretion of the immigration officer at your port of entry:

Quote:

Quoting Refugee or Asylee Travel Without Travel Documents

If you hold refugee or asylee status and are not a permanent resident, you must have a refugee travel document to return to the United States.

Derivative asylees and refugees must also obtain a refugee travel document before leaving the United States.

If you do not obtain a refugee travel document in advance of departure, you may be unable to re-enter the United States, or you may be placed in removal proceedings before an immigration judge.

I strongly suggest consulting an immigration lawyer with the specifics of your case.