Upvote:4
An asylum applicant can travel outside the US if advance parole is granted. See "what if I need to travel after I've applied for asylum" at https://www.uscis.gov/humanitarian/refugees-and-asylum/asylum/affirmative-asylum-frequently-asked-questions/questions-and-answers-affirmative-asylum-eligibility-and-applications:
If you applied for asylum and have not yet received a decision, you should not leave the United States without first obtaining advance parole. Advance parole allows certain individuals to return to the United States without a visa after traveling abroad. If you leave the United States without first obtaining advance parole, we will presume you abandoned your asylum application. Advance parole does not guarantee that you will be allowed to reenter the United States.
To obtain advance parole, you must file Form I-131, Application for Travel Document. For more information about travel documents, 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 (PDF, 45.16 KB).
Much has been made in the comments and in another answer of the fact that advance parole does not guarantee admission to the US. But even a green card does not guarantee admission to the US. This statement by itself is of little help in determining what might actually happen when the applicant returns to the US with advance parole. In fact, admission is likely unless the asylum application was denied while the traveler was outside the country.
Upvote:6
A Refugee Travel Document is only for people who have refugee or asylum status. You don't, yet, until your claim has been processed. You cannot "expedite" an RTD because you aren't eligible for it yet. You would need to wait until you have been granted refugee status before applying for an RTD.
Until then you either need to get Advanced Parole and use whatever travel documents you already possess or not travel.
If you have a lawyer handling your refugee application consult them about this.