The Deligne--Simpson problem is an existence problem for connections with specified local behavior. Almost all previous work on this problem has restricted attention to connections with regular or unramified singularities. Recently, the authors, together with Kulkarni and Matherne, formulated a version of the Deligne--Simpson problem where certain ramified singular points are allowed and solved it for the case of Coxeter connections, i.e., connections on the Riemann sphere with a maximally ramified singularity at zero and (possibly) an additional regular singular point at infinity. A certain matrix completion problem, which we call the Upper Nilpotent Completion Problem, plays a key role in our solution. This problem was solved by Krupnik and Leibman, but their work does not provide a practical way of constructing explicit matrix completions. Accordingly, our previous work does not give explicit Coxeter connections with specified singularities. In this paper, we provide a numerically stable and highly efficient algorithm for producing upper nilpotent completions of certain matrices that arise in the theory of Coxeter connections. Moreover, we show how the matrices generated by this algorithm can be used to provide explicit constructions of Coxeter connections with arbitrary unipotent monodromy in each case that such a connection exists.