Short answer first: CHOL requires the input matrix to be positive definite, it does not support positive semi-definite. I'll explain below why this is more practical for numerical computations.
Other things you can do instead of Cholesky decomposition:
[L, D, P] = ldl(A)
[U, d] = eig(A, 'vector');
d(d < 0) = 0;
[~, R] = qr(diag(sqrt(d))*U');
R = chol(A + 1e-13*eye(size(A)));
I hope one of these will do what you need.
Why CHOL doesn't (and shouldn't) work for semi-definite matrices
The problem is round-off error. When we numerically compute the eigenvalue decomposition, the eigenvalues and eigenvectors are those of a matrix that is very close to A, but not exactly the same: let's call it A_eig. The same happens when we compute the Cholesky decomposition, whose factors will be close to another matrix close that is close to A, let's call it A_chol. In your case, A_eig is just about positive definite, but A_chol is indefinite (positive and negative eigenvalues) - but for another matrix, it could be the other way around.
The errors A - A_chol and A - A_eig are guaranteed to be small, but they have a big impact for a matrix that is just barely positive definite. So for these matrices, some work-around is needed to reliably treat them as if they were positive semi-definite.