Consider particles on the integers or the real line that perform random walks and coalesce when they collide. The classical Karlin–McGregor theorem gives determinantal formulas for non-colliding particles on the line, but coalescence reduces the particle count and breaks the square matrix structure. Recently, Ákos Urbán extended the determinant to coalescing Pólya walks. We take a different approach: at each collision, a ghost particle continues along an independent path, preserving the total particle count and yielding determinantal formulas for any skip-free process at once. Summing out ghost positions recovers Urbán's formula. Applications include Rayleigh gap distributions, a Pfaffian point process structure for basin boundaries (extending Tribe–Zaboronski and Garrod–Poplavskyi–Tribe–Zaboronski), adaptation to annihilating systems, and a central limit theorem.