Why it can be counted
Each tile owns one of its edges, and the direction the path runs along it says which side of the path the tile is on. That is the inside/outside colouring above: two neighbouring tiles differ exactly when the path runs between them.
McKenna proved two facts that make the problem finite. A globally self-avoiding motif, laid on a torus by gluing opposite sides of the square, is a single loop through every lattice point (left: the motif and its eight neighbours, centre copy bright). And its whole outer ring of tiles is fixed by one of a few zipping modes, a palindromic string of bits with no two 1s together (shown up the left side). The bits say where the path steps off the outer track and back.
With the ring fixed, what remains is a planar puzzle: give every inner tile one edge so that the edges form a single path through every interior point. A transfer matrix counts those row by row without listing them. 11×11 takes a tenth of a second (McKenna's listing took 147 CPU-hours), 13×13 eleven seconds, 15×15 nine minutes.