EDIT. Walter and Trümper have announced on arXiv their implementation, with source code available, of two methods for testing total unimodularity. Their paper describes the technical details of the implementation / algorithm, and also provides several experimental results.
I found the following link for an implementation in R, where they claim to have a function for testing whether a matrix is totally unimodular. I have not checked which particular algorithm they use.
Link: R package
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