I understand that the wavefunction in Quantum Mechanics is usually treated as a complex vector with one real and one imaginary component. Does it make an actual difference in terms of the answers we get, if we treat the wavefunction as having two real components instead of one real and one imaginary component, or is it just a matter of convention to treat the wavefunction as having one real and one imaginary component instead of two real components?
There exists the general wavefunction , a solution of a wave differential equation which is a mathematical expression that allows solutions in real numbers and in imaginary numbers. Here you can see the particular differential equation and its solutions that was developed to make a theoretical physics proposal for the quantum mechanical ad hoc solution at the time, the Bohr model.
The wave equation developed by Erwin Schrodinger in 1926 shows some similarities in its one-dimensional form:
Note that the complex numbers are inherent in a quantum mechanical wave equation.
Physics theories use mathematics with its theorems and axioms as a tool, to develop calculations that will be descriptive and predictive, i.e. fit experiments and observations. To pick up the correct solutions from the infinity of mathematically possible ones, one uses new axiom like statements, called laws, postulates , principles, ever since the time of Newton (see link for postulates of quantum mechanics).
So by the postulates of quantum mechanics, the theory that describes the small dimensions, the only possible predictions are probabilistic. One can only predict the probability of measuring a particle at (x,y,z,t). This is the result of using complex numbers for the wave equation, which means that the wavefunction itself is not measurable, only $Ψ^*Ψ$ measurable , and gives the probability of finding a particle at (x,y,z,t).
So the answer is No, real functions as solutions of a wave equation cannot predict what the theoretical construct of quantum mechanics predicts for data.
To the observation by @mmesser that :
an imaginary number can be represented by two real numbers. You can turn the Schrodinger equation into two coupled real-valued equations.
my answer is that then extra postulates for quantum mechanics would be needed on how to determine the probability from two coupled differential equations. Complex algebra keeps the formats compact.