Householder QRD on complex matrices w/ the subcondition that the main diagonal of R has only real entires

Question

How to adjust the common Householder QRD algorithm so that the main diagonal of the R matrix has only real entries. Right now I use the algorithm described here https://arxiv.org/pdf/math-ph/0609050.pdf (page 19) to triangulize the matrix. I do not need the Q matrix, it should be used up on the vector that is attached to the matrix. I only need the (triangular) R matrix, with real entries on the main diagonal. Is there a common way to do this? Can this be done by rotating the i-th subrow, A[i][i,i+1,i+2...], (multiplying the subrow with $e^{(-i \theta)} : \theta = arg(A[i,i]))$ in i-th step of the algorithm and multiplying only i-th entry on the attached vector? If not, is there another way to do it? Maybe some literature... Thank you