  We consider determinants of Wronskian type whose entries are multiple
orthogonal polynomials associated with a path connecting two multi-indices. By
assuming that the weight functions form an algebraic Chebyshev (AT) system, we
show that the polynomials represented by the Wronskians keep a constant sign in
some cases, while in some other cases oscillatory behavior appears, which
generalizes classical results for orthogonal polynomials due to Karlin and
Szeg\H{o}. There are two applications of our results. The first application
arises from the observation that the $m$-th moment of the average
characteristic polynomials for multiple orthogonal polynomial ensembles can be
expressed as a Wronskian of the type II multiple orthogonal polynomials. Hence,
it is straightforward to obtain the distinct behavior of the moments for odd
and even $m$ in a special multiple orthogonal ensemble - the AT ensemble. As
the second application, we derive some Tur\'an type inequalities for multiple
Hermite and multiple Laguerre polynomials (of two kinds). Finally, we study
numerically the geometric configuration of zeros for the Wronskians of these
multiple orthogonal polynomials. We observe that the zeros have regular
configurations in the complex plane, which might be of independent interest.
