A certificate under 61.75 says "valid in conjunction with license 123456 from XYZ country." Or something similar.
If you SOLI from XYZ to ABC then the underlying license no longer exists.
Training for a stand alone private ASEL ticket under 61.109 requires visa/TSA/etc despite being "private-to-private." But, provides a permanent solution. Requires a knowledge test, "three hours training in the previous 60 days," and a checkride.