Stating mathematics is a tool doesn’t answer if mathematics are real or not. But I would say, from my humble experience, that mathematics is both unreal and perfectly tangible. Mathematics is totally a real thing as it obeys strict rule in logic that are true in our real world, axioms, on which everything else is based so that it can’t be used to state things as being true out of the blue, without any justification before using those axioms, which you can translate into our real world. But math also has its limits and has been used to demonstrate that it itself is incomplete, undecidable and inconsistant (mathematically, of course, it’s not our common definition here). Meaning, as mathematics are imperfect, it can’t describe our world perfectly and therefore isn’t real.

There is an excellent video from Veritasium on the subject of the limits of math: youtu.be/HeQX2HjkcNo