My standard answer (that we will never know and that's okay) is probably not what you're looking for? I think that the answer has to come in two parts when you follow the evidence. The first one is whether Richard was sleeping with men as just a regular part of his sexual expression. It is my opinion that the evidence does lead to this, but I've also seen arguments that I found compelling that Richard was simply not interested in women, which doesn't mean that he was interested in men and could just mean that sex was not a very high priority for him. I think most evidence indicates that women and men were both part of Richard's experience. Richard's immediate circle was his knights and trusted mercenaries. As far as I know, we don't have names for mistresses or ladies that he kept around, except for his mother. Unlike Philip, who put aside his wife to take up with another woman, Richard puts aside his wife, ignores all advice to get an heir ASAP, and doesn't seem to "take up" with anyone that could be politely recorded in chronicles. There is also the story of Richard's "vice", which I personally find hard to believe was women exclusively - as we only know of one child (maybe two, depending on your "nephew/son Henry" theory of choice) and because sex outside of marriage wasn't punished in this way for an unmarried king. There are stories of Richard's "lust for women" but they mostly seem to be in violent abduction and rape stories, which I think are different and come from a different place than genuine interest in women as a romantic object. There is a story that Richard demanding company on his deathbed, this elaborate orgy of women and his favorite companions, but I haven't found contemporary stories of this, only later 14th century stuff, but please correct me if I'm wrong.

Does this mean that Richard AND PHILIP specifically had a relationship? The story is better for sure if they did and I think it's our job as storytellers to explore that story. If you want to think of yourself as a historian, I think the burden is a little higher. The great quote that many use to imply Richard and Philip's relationship does (again, in my opinion) indicates more than just political schmoozing. Eating from the same plate and sleeping in the same bed is one thing - Henry II does it too - , but I do think the "loved him as his own soul" tagged on the end, is more telling than more conservative historians would like. I also think the extreme reaction that the two of them have to each other is more than just "you left me on Crusade" - refusing to be in the same room, rejecting communion because they hate the other person so much, sending John to bring messages because they won't see each other, trying to pay off the German emperor to keep Richard's ass in jail - these are all so forceful and feel really personal. John and Philip also had unfortunate political encounters, but they would still squirrel away and hash out treaties privately without all this dramatic show. And Philip's sex life wasn't something that should go without remark either - he was also a bit of weirdo and very extreme when it came to his personal, sexual relationships - see his wives, but also like.... Geoffrey maybe? I don't see why he wouldn't have been equally weird about a relationship with a man like Richard. I don't think it is without evidence to say that they may have had a little bit of situationship that was based on mutual personal and political interest. This isn't a YA romance novel - Richard wasn't trying to make them dual Kings of anything. Philip wasn't a politically naive uke who just wanted to be loved. If they slept together, it would surely have been with the full understanding of the crown ahead of the man. To extend the conversation a little further - Philip, I think, was very intrigued and excited by the Plantagenet boys as a whole - they lived a more boisterous life that he felt he couldn't have because of the Capetian ideal of sacral, steady monarchy with a very specific and important transition from father to son but I think he loved all four of them in their own turn and for different things. He was close with Henry and Geoff before their deaths, and I think there was some excitement in getting involved in Richard's succession crisis that also (if he played it right) would mean that he could extend France's reach. They had known (all of them) known each other their entire lives and that closeness is something that informs all the relationships between Philip and the Plantagenets. I think it's something that Philip regrets not having once John is dead. I think he struggles (and struggles to get Louis trained up) when it's just little Henry and a council that doesn't really know what their doing and who don't know him.