LIE women are rare, potentially the most rare type of all.
Add to that the tendency of Rational Duality being less common than Irrational Duality (based on my impression – and just think about it, do you know of more Rational or Irrational Dual pairs?),
it should be of no surprise that ♂ESI-♀LIE Duality is rare.
As far as I know, most ESI men end up dating a fellow ESI, sometimes an LSI, or an LSE once in a blue moon.
Many people in this thread have commented on male ESIs being more "grumpy" or "masculine". I don't find them any more or less grumpy than their female counterparts. In my experience, the Fi subtype of ESI is generally much more "grumpy" than the Se one. It could have something to do with the -Fi being strengthened. -Fi is not exactly known for "soft" and "accepting" moral judgment, rather the opposite. Also, while their Fi and by that their Fe is "boosted", I find that at the same time the modality of the function itself seems "boosted", too. In the case of ESI-Fi, their Fe seems to be "even more" Ignoring than the Fe of the Se subtype. But this is just my personal perception of the subtype differences.
Regarding the masculinity aspect – Gamma SFs can be fairly "masculine", as long as their Se is strong enough/boosted. I may be biased here as an Se-seeking female, but ESI-Fi men do not seem to fit into the stereotypical "masculine" gender role for the most part. Both Fi and Fe are usually seen to be more "feminine" than Te or Ti, hence an ESI-Fi does not seem to be that masculine, and can be rather androgynous in their gender expression (especially when they are Type 4). I feel like some people might mistake SLI men for ESI men. The former are the silent, logical, "grumpy" (because of Fe PoLR), practical, stereotypically "masculine" (because of being ST) type. ESI-Fi men, not so much. At last, I don't find ESI men rare at all, they are fairly common.
I cannot comment that much on ♂ESI-♀LIE Duality itself because I have not met nor observed a pair like that yet. I can only assume that their interaction would be fairly similar to the one between ♀ESI-♂LIE, with the exception that to onlookers their relationship will follow switched gender roles for the most part. I don't see LIE women being stereotypically "feminine", rather on the contrary. I have actually known one personally for a couple of years in Middle to High school. Just recently I realized she had been an LIE (likely Te sub) all along. We'd have frictions, needless to say. She was very vocal in Philosophy class and good at math. Known for being direct, straight-forward, loud, a bit klutzy (she is quite tall), and pretty smart. Before our interaction turned sour, she would indirectly ask me for dating advice with guys, or simply share how she had no luck with them. I cannot recall her dating anyone in Middle to High School, though she would have secret crushes on guys and not know how to properly act on them. She is currently studying industrial engineering. I have not seen nor talked to her since High school ended. And I honestly don't desire to.
I can give two other examples of what I deem to be female LIEs.
I am assuming those two ladies represent the two main archetypes of the female LIE you will come across.
Either the So/Sp or Sx/So version, essentially.
Here the real life example (likely So/Sp), Amy Webb:
And here the more glamorous, idealized fictional example.
I'd type her as LIE 3w4 Sx/So.
(I believe Jacob is LSI E6 and Shepard is SLE E8.)
Her reaction to the kiss seems pretty Victim to me.
And how she responds to his Aggressor advances before the kiss seems natural and positive.