I can't see anything in the rules which say it is not allowable to dance with two different partners in two different styles. Three if you want to add American Smooth. As Terence said . It is most unorthodox.
I'm one of those who believe that old saying. Jack of all trades . Master of none. I would bring to your attention one of the greatest Standard Dancers of all time Marcus and Karen Hilton.They were Latin Champions 1982 and 83 at Blackool long before they gave the Latin Style away and concentrated on the Standard Style. It wasn't untill they turned professional did they win a major title in the Standard Style.

I was in a similar situation (as your partner). My partner wanted to practice more and take more lessons than I was physically and financially able. He then said he would practice and take extra lessons with someone else. I felt he should be committed to our partnership and we broke up. Like your partner, I was OK with it at first but later changed my mind. It felt like an ultimatum to me, sort of a 'practice more, spend more, or I'll go elsewhere'. We danced worst than when we first started. So, since your partner agreed to add latin, but not more time and money, that means she can't do more. I don't think it is fair to her to do latin with someone else. Even though it is totally unrelated (in your mind), it may be a slap in the face to her.

Ok. This is a subject that I feel very strongly about. The absolute biggest reason you should not have plural partners is because when you do you dont become common with one person. Notice how the couples who are most tuned in to each other have been together for the longest time and only dance with each other. Everyone in this world has different tendencies and having one partner lets you get specifically tuned in to that persons natural tendencies. You should want to know the ins and outs of your partner! When you constantly switch partners your body doesnt get the chance to become naturally tuned in and loses focus. This is such a problem in Utah!

May I offer a contrary view, and one which is based on dance as a social form, and not in respect of dancesport.

No one who does not regularly seek out and enjoy dancing with a variety of partners can really dance at all. Dance is, fundamentally, a social activity, and restricting yourself to one partner is basically anti-social, and also means that instead of becoming 'attuned' to the nuances of your partner's style of movement, you lose all abiity to dance with good technique, and to develop, properly, the lead and follow skills that would enable any first-rate dancer to dance with anyone - complete novices included.

Having a preferred partner (spouse, commonly) and being particularly 'at ease' when dancing with them is one of the great pleasures of dancing, but dancing at our best with a regular partner is the product of our experience and skill, not the foundation for it.

I agree with Telemark that when dancing socially, it is important to have more than one partner. I dance socially every Friday night and dance with several men. However, the original poster was talking about competition. And in my experience, it is not possible to dance with more than one partner if you want to win. You need one coach and one partner. Concentrate on what your coach tells you and work with one partner to get it right. As soon as one partner feels inadequate or pressured, the partnership is doomed and the results will not be good. I also think that dancers should not forget that dancing is a social activity first and foremost. I'm always sorry when I hear someone say that they never dance socially anymore.

Totally agree with Ladydance.

Social - more the merrier! Never forget you were a new dancer once upon a time!

Comps - Can you really face more than one partner moaning?!

This is an example of the difference in attitudes between hobby dancers who may compete recreationally, and the traditions and decorum of more dedicated competitive efforts.

To some extent, you could call any competing professional who has pro/am students out on the inconsistency of this (and there are very real costs to their partnership that arrise from it) but for economic reasons that is seen as normal, while multiple partners is seen as a hobbyist thing to be outgrown if one wishes to really focus on achieving something in a partnership.