In general, an asylee who obtains a refugee travel document can leave and return to the United States. I recommend contacting a qualified immigration attorney to discuss the specific facts of your case to make sure there aren't any other issues that will prevent your entry back to the United States.
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There are two sides to this. First, you need to make sure there is no criminal issue that would prevent your return to the US. If there is not, then probably the fact that they approved the RTD makes it safe from the perspective of whether you will be allowed back in the US without problems.
However, a RTD is not a US passport. You might need a visa to enter a foreign country. That usually depends on your nationality. You should not travel on your foreign passport, however. You need to check with the consulates of the countries you intend to visit.Ask a similar question