What would be there instead if jealousy were absent?
There is no more unwelcome emotion inside the circus tent of feelings than jealously. Jealousy makes us feel shame for feeling it - we feel its nature is to fester and destroy, so obviously shameful to feel and even worse to enact it is. But every one of us feels jealousy at some point if we ever had the opportunity to desire something or someone strongly enough that we felt we'd be diminished by the lost opportunity or the absence of affections.
If an emotion is like an algorithm, it is a very compact way of signaling important information. We don't think through anger or love to get to the feeling - the feeling is present prior to the cognitive work. Every animal responds to emotions, and survives as well as it can due in large part to the information contained within emotion - flee, fight, protect, eat, desire.
We are in a zero sum game regarding others in some ways. There are lost opportunities, people we desire who do not desire us, and the fear of the loss of attention. So jealousy, though potentially destructive, is not a wrong emotion. What we need to think through is "what would this be like if jealousy were not present?" How would it be if your romantic partner felt absolutely no jealously when faced with the turning of your affections toward another? They would not be waiting up for you to return - there would be no anxiety when considering your permanent absence from their lives, no sense of loss, for there is always another potential partner to take your place. And if we turned it around, that might be just the way we'd feel (or not feel) too. This is not the life we want. As much as desire and attachment create the potential for jealousy, attachment and jealousy are two sides of a coin.
Step one is to acknowledge our jealousies. As petty and ridiculous as they seem, the shame of hiding them only makes things worse. If you can sit with the emotion of jealousy and really feel it, it is a complex of emotions, not just one. Fear of betrayal, loss of self-esteem, the recognition of some dependence upon another for our happiness, feelings of inadequacy regarding our romantic performance. These feelings all live together in the submarine that stays submerged as long as things are going well - but they are always there. This itself is a difficult pill to swallow - didn't our efforts at meditation and concentrated stretching flush all this stuff out? No.
Step two is to stop feeling shame about feeling jealous. You don't author your emotions, and you don't bring them under control by suppressing them. You might notice that by allowing yourself to feel the full cascade of jealousy (which is not the same as feeling it and then immediately planning how to manage the situation in order to escape having to feel it) there is a bit of daylight here and there in the turbulence of emotion. This is acceptance, actually. You accept that you are in fact, sometimes a jealous person and that when you feel jealously it is a part of you that is jealous, and that part need not be permanently removed because of your general distaste for it. You don't know in what way you'd be diminished by the removal of an integral part of your desires and the hopes for happiness, which jealously is.
Step three is to begin to admit to others, and particularly our intimate partners, that we can be jealous, and that we are as uncomfortable with it as they are when it appears, and that our attempts at control of our partner motivated by jealousy can be a barrier to expressing our real love for them and that we recognize that. To say we are "working on it" implies we know what to do about eradicating or diminishing our jealousy. We don't know what to do with our jealousy any more than we know how to stop being hungry, or tired, or desirous. What we can offer them is that we will take responsibility for our actions within the relationship, and offer apologies if our jealousy-driven actions hurts them. If you can imagine your partner sitting you down and proactively doing their best to explain their hopes for our continued presence in their lives and what of their behavior might interfere with that aim, it is hard to imagine being loved and respected more than that, harder yet to imagine a betrayal of faith to one so courageous.