What would remain of Quantum Computing if quantum states were real numbers?
The gate set Hadamard + Toffoli is universal for quantum computation [1] and only uses real number coefficients (some negative). So all quantum algorithms can be run without involving imaginary numbers.

That being said, you will pay some representation overhead if you stick with just H+Tof. For example, you need an extra ancilla qubit and essentially every encoded operation will act on this qubit. This creates a false dependency between most operations, preventing parallelism and locality.

[1]: https://arxiv.org/abs/quant-ph/0301040