(abridged) Observations of exoplanets indicate the existence of several correlations in the architecture of planetary systems. Exoplanets within a system tend to be of similar size and mass, evenly spaced, and are often ordered in size and mass. Small planets are frequently packed in tight configurations, while large planets often have wider orbital spacing. Together, these correlations are called the peas in a pod trends in the architecture of planetary systems. In this paper these trends are investigated in theoretically simulated planetary systems and compared with observations. Whether these correlations emerge from astrophysical processes or the detection biases of the transit method is examined. Synthetic planetary system were simulated using the Generation III Bern Model. KOBE, a new computer code, simulates the geometrical limitations of the transit method and applies the detection biases and completeness of the Kepler survey. This allows simulated planetary systems to be compared with observations. The architecture of synthetic planetary systems, observed via KOBE, show the peas in a pod trends in good agreement with observations. These correlations are also present in the theoretical underlying population, from the Bern Model, indicating that these trends are probably of astrophysical origin. The physical processes involved in planet formation are responsible for the emergence of evenly spaced planets with similar sizes and masses. The size--mass similarity trends are primordial and originate from the oligarchic growth of protoplanetary embryos and the uniform growth of planets at early times. Later stages in planet formation allows planets within a system to grow at different rates, thereby decreasing these correlations. The spacing and packing correlations are absent at early times and arise from dynamical interactions.
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