Burgers, chicken and hot dogs dominate the fast-growing, $684 million market for plant-based meat alternatives. Fish? Not so much.

That is starting to change as upstart companies invest in new ingredients, machinery and technologies that promise to make imitation tuna, salmon and shrimp smell, taste and appear more...fishy.

Heather Collins, a Denver, marketing consultant, was surprised by the vegan “smoked salmon” from Sophie’s Kitchen that she purchased a few months ago for a family gathering that included a vegetarian cousin.

“It had the smoked flavor, and it even had that slimy texture to it,” she says. The salmon never made it to the family dinner, Ms. Collins says. She ate it all herself.

Texture is the Holy Grail when it comes to finding plant-based alternatives that mimic meat, and fish has proved particularly difficult to get right.