Yes, you can eventually bring your spouse to the US, but I am sorry to inform you that this will take some time. You will need a green card or US Citizenship before you can apply for your spouse because the marriage happened after you received asylum status. I would recommend you apply for your spouse after you have received your green card as it will likely be a bit faster than waiting until you are a US Citizen.
As asylee is generally eligible to apply for a green card after 1 year in the US in asylum status. After you’ve applied it could take between 6 months to 1 year to receive your green card. Once you have the green card, you can then submit a petition to sponsor your foreign national spouse. Unfortunately, you then will need to wait another 1-2 years because a spouse of a green card holder is the F2A priority category, and this category is not current. See the visa bulletin for how long F2A category (spouses of permanent residents) are waiting.