Heat the oil in a fairly large saucepan, add the onions and cook them, with the lid on the pan, for about 10 minutes, until they are tender but not browned.
Add the garlic, the tomatoes together with their liquid, the basil and the wine.
Let the mixture simmer gently without a lid on the pan, stirring from time to time, until the liquid has disappeared and it is quite thick - this will take about 20 minutes.
Add the mushrooms and cook for a further 5 minutes or so, or until the mushrooms are tender and any liquid which they make has boiled away.
Remove from the heat and season to taste.
When the filling mixture has cooled, you can assemble the strudel, which can then be cooked straight away, or kept in the fridge for a few hours until you're ready, or in the freezer for at least a month.
Preheat the oven to 400.
If you have the type of phyllo pastry which is long and narrow - about 12 x 8 inches - lay two sheets side by side, overlapping them slightly where they join.
If you have the type which is quite large - about 12 inches or so square - lay out one sheet.
Either way, brush the surface with a little melted butter and then sprinkle with a third of the nuts.
Put another layer of phyllo pastry on top, brush with butter, and scatter with nuts.
Repeat with another layer, then a final layer of phyllo pastry, which you just brush with butter.
Now tip the tomato filling mixture on top and spread it to about 1 inch of the edges.
Fold the edges over, to enclose the edge of the filling then, starting from one of the widest edges, roll the whole thing up like a Swiss Roll, trying not to let it break.
Brush with melted butter and garnish with some shreds or shapes of phyllo pastry.
Put the strudel into the oven and bake for 30 minutes, or until golden brown.