Weird it is, certainly. I like to think of it backwards.
Mignini - corrupt prosecutor - has a murder on his hands. He is kicking his heels waiting for the evidence to come in - it doesn't happen overnight, despite what you might see on CSI. He sees the young couple who called it in. He starts thinking of his sex game thing. He has form for this. Then he starts bugging their calls, gets their phones etc - remember there is no forensic evidence at all at this point. They are interviewed as witnesses.
BINGO - a text to a black man saying "See you later" This is misinterpreted totally as an arrangement to meet up later. They have AK locked in a room with no lawyer and put it to her that she HAD planned to meet up with Lumumba later, after telling her that RS had disputed that she was indeed with him all evening (ie they asked if him if she COULD have left at any point and admitted, yes, well she could have gone out whilst I was asleep).
They tell AK they KNOW she was there with Lumumba and ask her to imagine what went on. She is totally scared, tired and confused and does in fact "imagine" something, In fact they keep on at her until she "imagines" the thing that suits their theory - at which point they write it all down. And arrest them all. The next day - both of them retract what they said. RS remembers later that AK couldn't have got out of his flat wthout the key, reinforcing his (probably correct) memory that she didn't leave.
2 weeks later they find evidence against Rudi Guede and arrest him. It turns out that Lumumba has a rock solid alibi so they have to let him go - despite attempts to turn it in to a 4-way crime. Evidence FIRMLY puts Guede at the scene. He admits he was there and says it was a strange Italian man that did the crime whilst he was on the toilet. That Meredith invited him in. There was NO mention of AK being present.
At this point the police SHOULD have rethought their theory - remember the bra-clasp evidence hadn't been found yet. BUT - they didn't. They kept with the sex story gone wrong - they had already released LOADS of stuff to the press, the bloody bathroom/footprints etc. Those "sinister" kissing photos were all over the DM. They were FIXED on their story - despite the available evidence not backing it up. And they did everything they could to make the evidence fit the theory - including claiming Meredith died much later in the evening as AK/RS COULD NOT have been present at the time she most likely did die, ie around 9pm.