Quasi-Hermitian quantum systems with a specified positive-definite physical metric and metric-compatible dynamics can be mapped instantaneously to equivalent Hermitian systems via a similarity transformation. Extending this equivalence consistently over a multidimensional parameter space requires compatibility among the proper similarity transformations obtained along different paths. We show that this compatibility can be characterized by a metric-induced connection. The curvature of this connection characterizes the local geometric obstruction to extending pathwise proper transformations to a common proper frame. Its holonomy, in turn, characterizes the global topological obstruction to single-valuedness that may occur even when the connection is flat. Finally, through three two-level models and a two-sublattice chain, we illustrate how the geometric and topological obstructions can be diagnosed in concrete systems, and demonstrate observable endpoint signatures arising from nontrivial holonomy together with possible simulation and readout protocols.