Access to Insight has an article titled A Happy Married Life: A Buddhist Perspective by Ven. K. Sri Dhammananda.
It's fairly detailed about many topics, but here's quoting one subsection from that:
Polygamy or Monogamy
To the question of whether Buddhists can keep more than one wife, the direct answer is not available in the Buddha's teaching, because as mentioned earlier, the Buddha did not lay down any religious laws with regard to married life although he has given valuable advice on how to lead a respectable married life.
Tradition, culture and the way of life as recognized by the majority of a particular country must also be considered when we practice certain things pertaining to our lives. Some religions say that a man can have only one wife whilst others say a man can have more than one wife.
Although the Buddha did not mention anything regarding the number of wives a man could have, he explicitly mentioned in His discourses that should a married man go to another woman out of wedlock, that could become the cause of his own downfall and he would have to face numerous other problems and disturbances.
The Buddha's way of teaching is just to explain the situation and the consequences. People can think for themselves as to why certain things are good and certain things are bad. The Buddha did not lay down rules about how many wives a man should or should not have which people are forced to follow. However, if the laws of a country stipulate that marriages must be monogamous, then such laws must be complied with, because the Buddha was explicit about His followers respecting the laws of a country, if those laws were beneficial to all.
I'm not sure what you mean by "romantic relationship". Maybe the Samajivina Sutta (AN 4.55) is an example:
As they were sitting there, Nakula's father said to the Blessed One: "Lord, ever since Nakula's mother as a young girl was brought to me [to be my wife] when I was just a young boy, I am not conscious of being unfaithful to her even in mind, much less in body. We want to see one another not only in the present life but also in the life to come."
And Nakula's mother said to the Blessed One: "Lord, ever since I as a young girl was brought to Nakula's father [to be his wife] when he was just a young boy, I am not conscious of being unfaithful to him even in mind, much less in body. We want to see one another not only in the present life but also in the life to come."
[The Blessed One said:] "If both husband & wife want to see one another not only in the present life but also in the life to come, they should be in tune [with each other] in conviction, in tune in virtue, in tune in generosity, and in tune in discernment.
I don't know but Polyandry in Tibet suggests that polyandrous marriage was practiced for socio-economic reasons (e.g. to avoid dividing inherited land among several separate families). I don't know whether that counts as "romantic".
I thought that maybe "romantic love" was a modern invention (and that traditional relationships had practical motives), but Wikipedia doesn't confirm or deny that theory. It does say,
... states that romantic love introduced the idea of a narrative to an individual's life, and telling a story is a root meaning of the term romance. According to Giddens, the rise of romantic love more or less coincided with the emergence of the novel. It was then that romantic love, associated with freedom and therefore the ideals of romantic love, created the ties between freedom and self-realization.
A "narrative" is, I'd guess, a "self-view" according to Buddhism.
I don't know much about polyamory. One little thing I read once, maybe not on-topic here, was that the Islamic Koran says something like, that polygamous marriage is OK as long as the husband treats all his wives equally -- and that some people interpret that as a prohibition because such equality is impossible in practice.