Is there a tape for joining cardboard that can be painted over afterwards?

If you want to stay with what you have, there is a workaround: tape the joint for structure and then cover it with a thin layer of paper and PVA glue.

A strip of ordinary paper, or a piece of tissue for a nearly invisible edge, glued over the taped seam gives you a paintable surface. It is the papier-mâché approach and it works on any joint regardless of what is underneath.

More steps than the gummed tape, and useful when the tape is already on and you would rather not rebuild.

One more option for corrugated specifically: for a really clean edge, mitre the joint and glue it with PVA, then reinforce from inside with tape where nobody will see it. The outside is then bare cardboard everywhere.

There is exactly the tape you want and it has been used for a century: gummed paper tape, the water-activated kind used for sealing cartons.

It is kraft paper with a dry starch adhesive on one side. You wet it, apply it, and it bonds into the surface of the cardboard as it dries rather than sitting on top of it. Once dry it is paper — it takes paint, gesso, glue and further layers exactly as the cardboard does, and the joint disappears under a coat of primer.

The reinforced version, with fibreglass threads embedded between two paper layers, is considerably stronger than plastic packing tape and still fully paintable.

This is also the traditional material for the same job in scenery and papier-mâché work, which is a good sign that it holds up.

One technique note: wet it enough to activate the adhesive and not so much that the paper goes limp. A damp sponge in a shallow tray is the usual method.

The framing-supply version of the same product is worth knowing about: gummed linen or paper hinging tape, sold for mounting artwork.

It is more expensive and better made — finer paper, more consistent adhesive, and designed to be archival. For small models where the joint quality matters more than the cost, it gives a much neater result than carton tape, which is fairly coarse.

Either way, the general principle is the one to hold onto: for a joint that will be finished like paper, use a paper-based joint. Any plastic film will always be a different material under the paint, and no primer entirely hides that difference in sheen and texture.

Whatever joint you use, seal the whole model before painting.

Corrugated cardboard drinks water-based paint unevenly and the surface texture and joins show through. A coat of dilute PVA or a proper primer over everything makes the surface uniform, and after that a paper-taped joint and the panel beside it genuinely look the same.

Skipping this is the usual reason a well-made model still looks patchy.