Gutter can be clear, but can be overwhelmed by the amount of rain and the gradient of the guttering to take it away.
My guess is going down the gaps of the verge caps and down/behind the fascia, they are not broken. Expect those gaps all the way up, so problem could start higher up.
I've no experience with your style of roof tiles, so can't suggest a fix. I prefer to have a mortar verge, not capped.