Hi Brian,
I was told that a same-sex couple recently obtained their residency visas with one partner as a dependant by presenting their U.S. marriage certificate here in Ecuador, so I think it is doable. However, this information is second-hand and I don’t know the couple personally so I can’t verify the story.
But, I would suggest contacting a lawyer here to get the most up to date visa regulations and requirements. On the home page of the Ecuador Insider you’ll find the Rolodex section on the bottom right of the page. In there is a nice list of lawyers in Ecuador and all of their contact info. One of them should be able to give you a definite answer on this.