Opposite-sex（主语）// are（系动词） couples（表语）// who are eligible to marry and may have their  marriage performed in any state and have it recognized in every other state in the nation and every country in the world（修饰表语的定语从句）.

异性是指有资格结婚的夫妇，他们的婚姻可以在任何州进行，并且在这个国家和世界上的每个其他州都得到承认。

备注：以上句子情态动词may前面要加上一个连词and，否则句子是一个错误句子。

加了连词and后，定语从句有以下3个谓语动词。

(1)are eligible to marry

(2)may have their marriage performed...

(3)have it recognized...

have their  marriage performed（属于have sth done 结构）

have it recognized （属于have sth done 结构）

Opposite-sex are couples who are eligible to marry and may have their  marriage performed in any state and have it recognized in every other state in the nation and every country in the world.该句may前面漏了一个连词and。参考译文：符合结婚条件的一对配偶其性别是相异的。他们可以在本国的任何一个州登记结婚、举行婚礼，他们的婚姻在本国任何其他的州以及全世界任何国家都会得到认可。