Stop trumpet vine from growing under my deck between the wood boards

I have a 10' x 20' wood patio, about 1 foot off the ground, with a set of tell aces on either end, gorgeous trumpet vines growing up on them. The issue is, the trumpet vines send underground rhizomes and every week from May through September (I'm in Michigan), we're out there pulling the vines up through the duck's wood boards, clipping away as much as we can. A week later it's all back again. If we were to leave it 2 weeks, it's a jungle!

I'm afraid to use round up, as wouldn't it have the possibility of killing the main vine, which I don't want to do?

Crawling under the deck isnt an option, not enough space. But maybe a roll of landscape tarp?

Or is manually clipping them every week going to be what I need to do?

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