The idea that I have heard about the personality (according to the Big Five) and romantic partner is that the best combination are two similar people, but not the same. That the differences in personality are at the same time the part that lets people grow and the part that bring problems in the relationship.
For example, the difference in extroversion can bring forth the problem of how many things will they do together and how social will the things they do together be. Because the introvert would rather be in a small group or one-on-one or even have some time alone for themselves. But the extrovert would most likely prefer going out all the time and meeting zillions other people.
The difference in agreeableness could be show in the difference of how much do people adapt to other people. The agreeable people let other people make up their mind about most if not all things. And they have a way to finish the discussion with the words: "whatever you want". On the other hand, the disagreeable people are much more likely to make their own mind, and if or when they do, they will tell it. So they will find the "whenever you want" approach annoying.
The difference between conscientiousness would be shown in the similar matter to the workaholics vs. relaxed person. The workaholic would be the one always working and then complained that the other person never does anything. On the other hand, the person on the other side of the spectrum would complain that they can never relax and it is always work, work, work, and for the change they should just let it go and do something fun.
The difference between the neuroticism could be shown for example in the financial dealing. The person with the higher neuroticism would most likely wanted to have savings, while the person with low neuroticism would not. Or how much to check something before leaving or what they are actually willing to do and so on. Now that I think about, the person with high neuroticism would want savings, but a person with high consciousnesses is much more likely to actually create it.
And the last, the difference in openness could be shown in what could be discussed. And also in how many new things the people would try. So one low on openness could consider the person high on openness crazy and unreasonable, as they would always want to do something new, change something or have a non-conventional idea and wanted to discuss it. On the other side, high openness people might see low openness people as boring, bland and too conventional.
Well, if I would be more of romantic or high on agreeableness, I the ending would have more emphasis on the partner, but I am neither, so this two questions are the good ending for me. Ask yourself: which difference would you be most willing to live with? And which part of yourself do you want to change?