There is a purely algebraic characterisation of Condition (B) due to Le and Teissier, see Proposition 1.3.8 of the paper

<cite authors="Lê Dũng Tráng; Teissier, Bernard">_Lê Dũng Tráng; Teissier, Bernard_, [**Limites d’espaces tangents en géométrie analytique. (Limits of tangent spaces in analytic geometry)**](http://dx.doi.org/10.1007/BF02566778), Comment. Math. Helv. 63, No. 4, 540-578 (1988). [ZBL0658.32010](https://zbmath.org/?q=an:0658.32010).</cite>. 

Here's a summary. Assume that $X$ is a complex algebraic variety (embedded in $\mathbb{C}^n$ if affine or $\mathbb{P}^n$ if projective) and that $Y \subset X$ is a smooth quasiprojective subvariety. Then the pair $(X_\text{reg},Y)$ satisfies Condition (B) if and only if there is a containment of ideals $$I[\textbf{Con}(X) \cap \textbf{Con}(Y)] \subset \overline{I}[\kappa_X^{-1}(Y)]$$

Some explanation: here $I[Z]$ means the generating ideal of $Z$, while  $\textbf{Con}(X)$ is the conormal variety of $X$ and $\kappa_X:\textbf{Con}(X) \to X$ is the conormal map. The bar on the right side here denotes *integral closure*. I learned about all this from Chapter 4 of Flores and Teissier's amazing survey

<cite authors="Flores, Arturo Giles; Teissier, Bernard">_Flores, Arturo Giles; Teissier, Bernard_, [**Local polar varieties in the geometric study of singularities**](http://dx.doi.org/10.5802/afst.1582), Ann. Fac. Sci. Toulouse, Math. (6) 27, No. 4, 679-775 (2018). [ZBL1409.14002](https://zbmath.org/?q=an:1409.14002).</cite>

Martin Helmer and I have [a recent paper][1] which uses this Le-Teissier criterion to algorithmically construct Whitney stratifications of complex varieties. The hard part is bypassing the integral closure, which is computationally prohibitive. Martin even has [a Macaulay2 implementation][2] on his webpage which you can play around with if you have actual example varieties to stratify :)



**Update (11th July 2023)** There is an error in my paper with Martin mentioned above, which has been described and corrected in [this erratum][3]; the fix involves using primary decompositions rather than saturations. The [arxiv version][4] incorporates all of the corrections.


  [1]: https://link.springer.com/article/10.1007/s10208-022-09574-8
  [2]: http://martin-helmer.com/Software/WhitStrat/
  [3]: https://link.springer.com/article/10.1007/s10208-022-09602-7
  [4]: https://arxiv.org/abs/2106.14555