I'm trying to think of examples but failing.

Imagine a romantic story set in our world, but there are some secrets in the plot. No vampires or anything like that, but a bit of mystery, intrigue, so that the main characters have to uncover secrets and stuff.
Oh, I know. Arisa That's the kind of manga I want. It can have less mystery and more romance, that's okay. But I want that feeling of suspense yet without any monsters or ghosts along with a developing romantic story.

Thanks in advance!

Cheese in the Trap!!!

I'm not really sure if this is up your alley but Yuureitou is a fave of mine.

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Biblia Koshodou no Jiken Techou
Gosick
Doubt! (AMANO Sakuya) <-------- this one

Teru Teru x Shounen

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I second Cheese in the Trap
Piece for sure.
Kechonpa
Torikago no Tsugai
Coelacanth
Korosareru nara, Isso Sakura no Ki no Shita de
Lady Detective if you don't mind historical.

Qualia the purple comes to mind, set in our world but with supernatural sci-fi elements.

Boku dake ga Inai Machi MC can repeat events in the past, but his control over the power is limited. All of his attempt to help people lands him in either a trauma, or a lot of trouble (he's currently ... yeah). Nothing even went right in his life, except for one girl who appreciates his endeavor

Spirit Circle from the author of Hoshi no Samidare (if you don't know what it is, then you never know romance in mysterious plot lol), tells a story about pair of boy and girl who are reincarnated over and over again to hurt each other. Very moving story about love, relationship, and regret.

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My list (rated by addictiveness, not necessarily quality)
Reading
Complete

I third Cheese in the Trap (amazing story-telling by the way)
Second [m]Teru x Teru Shounen[/m] (wonderful mysterious pasts, especially recently!)
Second Kechonpa (just cause I like the main character)

Now onward to the suggestions
Detective Conan ?
Kodomo no Omocha sorta
Kumo no Mukou, Yakusoku no Basho but there's no concrete ending.. and its not modern day..
Keishichou Tokuhanka 007 some supernatural stuff
Penguin Brothers i think...
Spiral: Suiri no Kizuna gets confusing and wordy towards the end... but still good

Utsuro no Hako to Zero no Maria (Novel)
Melo Holic
Gekkou (Novel)

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Not sure if you'd like,
try Another-In the spring of 1998, Kouichi Sakakibara transfers to Yomiyama North Middle School. In class, he develops a sense of unease as he notices that the people around him act like they’re walking on eggshells, and students and teachers alike seem frightened. As a chain of horrific deaths begin to unfold around him, he comes to discover that he has been placed in the cursed Class 3 in which the student body head count is always one more than expected. Class 3 is haunted by a vengeful spirit responsible for gruesome deaths in an effort to satisfy its spite. To stop the vicious cycle gripping his new school, Kouichi decides to get to the bottom of the curse, but is he prepared for the horror that lies ahead…?

With spirits but Great mystery manga though

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Tournament Round 1
Round 2

Thank you!
I welcome more recs ^___^

Oh, and if you know some anime like this, add the title, too, please.

I fourth Cheese in the Trap

Hm.. Well there's Shinrei Tantei Yakumo. It's got a couple of supernatural aspects to it, but it's not like full on, right in your face, kinda thing. I personally like it, and the art is nice and clean :3

There's also Hyouka, however there's only 4 chapters out right now. But I'd recommend watching the anime if you don't mind. It's also really good

Hope I helped