There are many hidden reinforcements you could use - biscuits, dowels, or more complex hidden joints such as these:
Mitred corner dovetail joint Mitred blind dovetail joint ...
but I wouldn't recommend them unless you have a lot of skill with a chisel, and a lot of time on your hands.
Both of these images are from here which is a really excellent site for all kinds of woodworking terms, and has lots of variations on common joints etc.
woodworking.stackexchange.com/users/272/WhatEvil
How might this joint be constructed so that it is strong enough to prevent racking but still appear to be a miter?
It's not racking that is the major concern here
, it would be (probably sudden!)
failure of the glue line in one or both of the mitres.
The answers from WhatEvil and Matt cover many of the solutions.
There are a number of variations on blind dovetailed mitres, but the others are of increasing complexity and hence harder and harder to make accurately.
To add some details re.
splined mitres (a stopped splined mitre to be specific, also called a hidden spline mitre joint or blind splined mitre joint): With the bench being made from fairly beefy material
it allows for quite a substantial spline being used, as I've illustrated above.
Even made from hardwood (with the grain oriented correctly, as shown) it is probably strong enough to be impossible to break in normal use, if plywood is used it would be stronger still.
Almost certainly overkill but for absolute assurance the spline could be made from metal (in which case it should be glued in place with epoxy or polyurethane adhesive).
Bonus joint: if a slight difference in appearance would be acceptable the rebated or rebbeted mitre joint could be used as well:
This is a significantly strong joint and may be the easiest wood-only solution as only three saw cuts are required per edge.
woodworking.stackexchange.com/users/243/Graphus
I've thought about this some more and have come up with an alternative solution.
If I had to make this bench and be sure that the joints won't break over time, I'd want to look at some real reinforcement...
how about a steel substructure?
So you have your basic bench structure which is made by lamination of smaller timber boards:
The fact that it's made of smaller timber boards allows you to sandwich the boards around a bent steel plate: Cross section of the whole bench: Cross section through main part of bench showing steel housed within timbers, with optional dowel: Fully exploded view:
So basically: Get 2 pieces of steel cut and folded (which should be quite cheap from a local fabricator), I have drawn them at 5mm thick with a 60mm timber thickness.
Get your constituent timber pieces for lamination and groove them out using a router or a circular saw.
I have shown a 10mm groove width to allow tolerance for the steel, though in reality the less tolerance you can have the better.
Optionally, drill dowel holes to aid in positioning / holding the whole thing together.
Glue
the whole timber part of the bench up around the steels.
I'd advise either using an epoxy glue to glue the steel into the timber and take up any tolerance, or wedge the steel tightly into the timber using timber wedges or maybe standard plastic packers or wedges, but you can get away with using PVA wood glue for the timber itself.
Now you have a really sturdy bench that looks simply constructed, and isn't actually that hard to do.
If you build it right, this thing isn't breaking in a hundred years.
Bear in mind
this is just a concept.
You could use just one central steel, or 3 steels, or smaller, separate steel brackets at each end, or you could use biscuits to reinforce between the horizontals either side of the grooves.
The basic premise is the important thing and that's to build the laminations up around steel brackets in order to add hidden reinforcement.
woodworking.stackexchange.com/users/272/WhatEvil
While I may (justifiably) be shunned for bringing it up, it is possible to get right-angled dowels .
http://www.fastenersplus.com/90-degree-right-angle-furniture-dowel-pkg-3.html
This will prevent any motion of the pieces, but it's not what you call a purist approach to woodworking.
woodworking.stackexchange.com/users/463/WhatRoughBeast
Metal Brace
This is not my area of specialty at all
but if I had to guess I would say that there are some metal brackets hiding underneath the joins that we cannot see.
Quite possibly in conjunction with what ever joint is hiding in there.
Just as long as they are a strong metal.
Not like the cheap ones made that are zinc plated you commonly find in hardware stores.
They will bend a twist without much effort.
Splined Mitre
I don't see this with a bench
but you could hide a large splinted mitre, that does not transverse the entire width, reinforced with screws or bolts underneath.
Graphus has the correct terminology for this joint: a stopped splined mitre to be specific, also called
a hidden spline mitre joint or blind splined mitre
joint
I would be wary not using some metal hardware to reinforce this bench.
Most people don't lightly rest the bottoms on a bench.
woodworking.stackexchange.com/users/128/Matt
Besides mitered versions of various types of other sufficiently strong joinery methods, you could use non-mitered versions of any of those, and more (dovetail, dowels, Domino, mortise and tenon, box joints, dadoed/housed joints, knock-down hardware, etc.--perhaps even butt joints with countersunk lag screws), using any sufficiently strong material.
Then, when you're all done, veneer the whole thing
so it looks like you used miter joints.
This wouldn't be my first choice, but the parts of the bench look thick enough that almost any material would be strong enough.
You could certainly achieve the same look with inexpensive materials, artificially-distressed veneers (even 1/8" or thinner might do it), and faux miter joints.
woodworking.stackexchange.com/users/49/rob
I own the bench along with the matching table.
It is a simple frame constructed with 2x3 dimension lumber; a simple reclaimed board veneer with miters.
woodworking.stackexchange.com/users/1480/dale
In this bench, the joints appear to be simple miters; however, I can't believe that this piece would be stable with a simple miter. How might this joint be constructed so that it is strong enough to prevent racking but still appear to be a miter?
What might be stronger than a miter but still look like a miter joint?
woodworking.stackexchange.com
<joinery><racking>
woodworking.stackexchange.com/questions/1453
woodworking.stackexchange.com/users/180/Daniel B.
Bonus joint, metal brace, splined mitre may all look like a miter joint. Besides mitered versions of various types of other sufficiently strong joinery methods, you could use non-mitered versions of any of those, and more (dovetail, dowels, Domino, mortise and tenon, box joints, dadoed/housed joints, knock-down hardware, etc.--perhaps even butt joints with countersunk lag screws), using any sufficiently strong material.   Besides mitered versions of various types of other sufficiently strong joinery methods, you could use non-mitered versions of any of those, and more (dovetail, dowels, Domino, mortise and tenon, box joints, dadoed/housed joints, knock-down hardware, etc.--perhaps even butt joints with countersunk lag screws), using any sufficiently strong material. rebated or rebbeted mitre joints could be used. Right-angled dowels, splined mitres and metal braces are other alternatives.
Rebated or rabbeted mitre joints could be used It is possible to get  right-angled dowel Metal brace Splined mitre Besides mitered versions of various types of other sufficiently strong joinery methods, you could use non-mitered versions of any of those, and more (dovetail, dowels, Domino, mortise and tenon, box joints, dadoed/housed joints, knock-down hardware, etc.--perhaps even butt joints with countersunk lag screws), using any sufficiently strong material. 