Greetings Nancy,

There are only a couple options for ridding your garden of humufus lupulus Aureus other than a glyphosate or triclopyr herbicide.

If you are able to remove the largest rhizome crowns by digging them out, that will slow the spread. Note that it may be easier to remove the crowns in fall. And be sure to put them in the trash as they will re-sprout in a compost pile. Once the larger pieces are removed, cover the affected area with landscaping cloth or thick cardboard. Any sprouts that escape the covered area should be promptly pulled to deprive the remaining roots of energy. It will likely take several seasons to completely remove it. 

Even just cutting back the vines to the ground repeatedly without removing and/or covering the crowns will gradually weaken the plant over time.

In many ways, humufus lupulus removal is similar to English Ivy or Himalayan blackberries and you may find some other helpful tips in these articles:

https://extension.oregonstate.edu/catalog/em-9710-ivy-removal-home-landscape
https://extension.oregonstate.edu/news/tackling-invasive-himalayan-blackberry-takes-persistence-planning

Happy gardening,