The steering vector basically applies the delay according to the spatial location of the Uniform Linear Array (ULA).
The multiplication by the complex exponential basically applies the delay.
The answer to your question is the Hilbert Transform.
You may use it to transform the data into Analytic Signal and then apply the algorithm.
For instance, I took 3 signals as in your question (Different phase) and set them a different angle of arrival.
I calculated the signal on each element of the array and applied MUSIC to get:
As you may see the algorithm found the spatial angle quiet easily (Well, this is a basic simulation).
Applying the same process just without the Hilbert Transform yield:
As one can see while the results are still accurate they are different.
The reason for that is for a different, and interesting on its own, question.
But it shows that there is no issue running the MUSIC algorithm on real signals.
The code is available at my StackExchange Codes Signal Processing Q82918 GitHub Repository (Look at the
Remark 001: I recommend reading Marcus Miller's answer to the question DOA - 1D Music Algorithm.
Remark 002: Another motivation for using "Real Valued MUSIC" is computation. So there are method that even in the case of a complex signal, the covariance is decomposed into Real Valued matrices in order to make the computation more efficient.