Lisbon—Mamadu Indjai has given up on Europe and is heading home to the former Portuguese colony of Guinea-Bissau. The 55-year-old West African has spent the past 19 years in Portugal. Yet all he has to show for his labors are the house he was building for his family back in his ancestral village of Caio. “I haven’t got the strength to struggle anymore,” Indjai sighs.

Efforts to create a hospitable Iberian Model for immigrants, largely from former colonies of Spain and Portugal, have been undermined by economic catastrophe, which has drastically reduced or eliminated budgets for integration and social services and forced many immigrants to leave for their homelands or elsewhere. It’s a dramatic reversal of an earlier and promising pattern. At the very moment other Europeans had all but closed their gates to what was seen as creeping Islamization, Muslim immigrants discovered the Iberian Dream. Barely a decade ago,...

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