Having a Green Card through marriage generally allows you to travel freely in and out of the United States. However, if you have an open pending asylum case, traveling back to your home country could have serious consequences for your asylum case and potentially your Green Card status.

Potential Consequences:

• Undermining your asylum claim: Traveling back to your home country could be seen as evidence that you do not have a well-founded fear of persecution there, potentially leading to the denial of your asylum case.
• Abandonment of your application: Returning to your home country could be interpreted as abandoning your asylum application.
• Scrutiny of your marriage: It might raise suspicions about the legitimacy of your marriage, leading to further investigation and potential revocation of your Green Card.
• Inadmissibility issues: If your asylum case is denied after you return, you could face inadmissibility issues and potential deportation.

If you withdraw your asylum application before traveling:

The risks are significantly reduced, but not entirely eliminated. The fact that you had an asylum application and then withdrew it remains on record and could raise questions in future interactions with immigration authorities.

However, having a Green Card is a valid reason to withdraw your asylum application. Withdrawing under these circumstances would likely be seen as reasonable and responsible. It's still important to consult with an immigration attorney to ensure the withdrawal is done properly and documented correctly.

Recommendation: It is strongly advised that you do not travel back to your home country while your asylum case is still pending, even if you have a Green Card through marriage.

For personalized advice, contact me for a consultation or contact a qualified immigration attorney.

Disclaimer: This response is general information and does not constitute legal advice. I am an immigration attorney, but I'm not your attorney until an attorney-client relationship is established.