I found there is no exact word to translate this phrase. Is there a way to describe this status?
曖昧 = ambiguous romantic relationship
This phrase is in reference to a romantic relationship. Its usually from outsiders commenting on the relationship between a guy and a girl who are friends but may be more than that.
You can kind of thinks about it as the sort of weird stage before you've had the DTR (determine-the-relationship) conversation. If you don't have that convo, then you may be perpetually 曖昧.
As many others have said, 暧昧 can be used in different occasions, which calls for different word to express the precise meaning in each occasion.
there's the romantic 暧昧 when you describe a relationship, which I would translate to something to the effect of "there's a thing between them".
when you talk about someone's attitude is 暧昧, you mean they are being indirect, noncommittal, giving mixed signals, hesitating to express their opinions. You would choose the one that fits the situation.
I think in general when you have a word that doesn't have a direct, 100% corresponding word, it most likely falls to your subjective judgement. And sometimes you simply don't have sufficient information or context to translate it, 暧昧 can be one of such word, for those occasions you would try your best and note what word is being used originally or ask for clarification.
Google translate gives it as the following adjectives:
暧昧, 含糊, 含混, 双关
dubious 可疑, 暧昧, 不三不四, 怀疑的
equivocal 暧昧, 双关
doubtful 疑, 儗, 暧昧, 踯, 怀疑的
shady 阴凉, 荫, 背阴, 不三不四, 暧昧, 荟
indefinite 不定, 活络, 暧昧, 暗昧, 不明确, 非决定性的
uncertain 不确定, 飘忽, 渺茫, 疑, 易变, 暧昧
indistinct 模糊, 缈, 恍, 惚, 苍莽, 朦
Given the translated terms, I'd add "vague" to the list.
Given the commentary below, perhaps the best fit might be the phrase and usage popularised by Facebook: "It's complicated"